M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2024
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
In-person direct teaching
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
doc. RNDr. Martin Čadek, CSc. (seminar tutor)
Guaranteed by
doc. Lukáš Vokřínek, PhD.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 14:00–15:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Wed 10:00–11:50 M2,01021, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two tests at the tutorials. It is required to obtain in total more than half of the points from these tests. In addition, the maximum allowed number of absences at the tutorials is set to 3 (in the case of the distance education, the presence will be supplemented by working out homeworks).
The final exam consists of a written and an oral part. It is required to obtain at least 50% of points from the written part. For the colloquium type completion of the course, students will only sit the written part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~koren
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2023
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
doc. Lukáš Vokřínek, PhD. (lecturer)
Joanna Ko, M.Sc. (seminar tutor)
Guaranteed by
doc. Lukáš Vokřínek, PhD.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 8:00–9:50 M6,01011
  • Timetable of Seminar Groups:
M3130/01: Tue 18:00–19:50 M6,01011, J. Ko
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two tests at the tutorials. It is required to obtain in total more than half of the points from these tests. In addition, the maximum allowed number of absences at the tutorials is set to 3 (in the case of the distance education, the presence will be supplemented by working out homeworks).
The final exam consists of a written and an oral part. It is required to obtain at least 50% of points from the written part. For the colloquium type completion of the course, students will only sit the written part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~koren
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2022
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Joanna Ko, M.Sc. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 14:00–15:50 M3,01023
  • Timetable of Seminar Groups:
M3130/01: Fri 8:00–9:50 M6,01011, J. Ko
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two tests at the tutorials. It is required to obtain in total more than half of the points from these tests. In addition, the maximum allowed number of absences at the tutorials is set to 3 (in the case of the distance education, the presence will be supplemented by working out homeworks).
The final exam consists of a written and an oral part. It is required to obtain at least 50% of points from the written part. For the colloquium type completion of the course, students will only sit the written part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~koren
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
autumn 2021
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 M5,01013
  • Timetable of Seminar Groups:
M3130/01: Wed 14:00–15:50 M2,01021, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two tests at the tutorials. It is required to obtain in total more than half of the points from these tests. In addition, the maximum allowed number of absences at the tutorials is set to 3 (in the case of the distance education, the presence will be supplemented by working out homeworks).
The final exam consists of a written and an oral part. It is required to obtain at least 50% of points from the written part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~koren
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2020
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 8:00–9:50 M6,01011
  • Timetable of Seminar Groups:
M3130/01: Wed 14:00–15:50 M4,01024, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. In addition to having max 3 absences at the tutorials (the presence will be replaced by handing in homeworks in the case of the distance tutorials), it is required to obtain in total more than half of the points from these tests.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~koren
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2019
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 10:00–11:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Tue 12:00–13:50 M2,01021, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- quadrics and their classification,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.
Learning outcomes
At the end of this course students will be able to:
- understand the connection between bilinear forms and the geometry of quadrics;
- compute invariants of quadrics and derive their geometric properties;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. In addition to having an allowed number of absences at the tutorials, it is required to obtain in total more than half of the points from these tests.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2018
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
doc. RNDr. Martin Čadek, CSc. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 9. to Fri 14. 12. Mon 10:00–11:50 M2,01021
  • Timetable of Seminar Groups:
M3130/01: Mon 17. 9. to Fri 14. 12. Tue 14:00–15:50 M4,01024, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
autumn 2017
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 18. 9. to Fri 15. 12. Mon 8:00–9:50 M3,01023
  • Timetable of Seminar Groups:
M3130/01: Mon 18. 9. to Fri 15. 12. Tue 14:00–15:50 M4,01024, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2016
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 19. 9. to Sun 18. 12. Mon 14:00–15:50 M2,01021
  • Timetable of Seminar Groups:
M3130/01: Mon 19. 9. to Sun 18. 12. Tue 14:00–15:50 M1,01017, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2015
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 13:00–14:50 M4,01024
  • Timetable of Seminar Groups:
M3130/01: Wed 15:00–16:50 M4,01024, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2014
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 12:00–13:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Wed 12:00–13:50 M6,01011, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2013
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 12:00–13:50 M2,01021
  • Timetable of Seminar Groups:
M3130/01: Wed 14:00–15:50 M3,01023, L. Vokřínek
M3130/02: Wed 12:00–13:50 M3,01023, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2012
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 12:00–13:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Tue 8:00–9:50 M5,01013, L. Vokřínek
M3130/02: Wed 14:00–15:50 M6,01011, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2011
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Tue 14:00–15:50 M4,01024, L. Vokřínek
M3130/02: Wed 16:00–17:50 M5,01013, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, projective closure, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Selected applications: spectral decomposition, Moore-Penrose pseudoinverse, Markov chains
  • Multilinear algebra: dual space, tensor product, exterior and symmetric products, tensor coordinates, functor Hom and its relation to the tensor product.
  • Integer matrices: equivalence, Smith normal form, classification of finitely generated commutative groups
  • Polynomial matrices: equivalence, Smith normal form, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
  • LANG, Serge. Linear Algebra. Third Edition. New York: Springer-Verlag, 1987, 296 pp. ISBN 0-387-96412-6. info
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2010
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 12:00–13:50 M5,01013
  • Timetable of Seminar Groups:
M3130/01: Wed 16:00–17:50 M1,01017, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2009
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 12:00–13:50 M5,01013
  • Timetable of Seminar Groups:
M3130/01: Fri 10:00–11:50 M5,01013, L. Vokřínek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2008
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 15:00–16:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Thu 16:00–17:50 M1,01017, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations. At the end of this course, students should be able to understand connections between bilinear forms and the geometry of quadrics, to know basics of multilinear algebra and to compute the Jordan canonical form of a matrix via characteristic matrix.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods
Form: lectures and exercises. Exam: written and oral Requirements: to know basic theory from the lectures, to obtain internal credit from exarcises.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2007
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 N41
  • Timetable of Seminar Groups:
M3130/01: Thu 8:00–9:50 N41, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2006
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
doc. Mgr. Jaroslav Hrdina, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Timetable
Mon 8:00–9:50 N21
  • Timetable of Seminar Groups:
M3130/01: Thu 10:00–11:50 UP2
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2005
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
doc. Mgr. Jaroslav Hrdina, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Timetable
Tue 13:00–14:50 UP2
  • Timetable of Seminar Groups:
M3130/01: Thu 8:00–9:50 UP1, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2004
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
doc. Mgr. Jaroslav Hrdina, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Timetable
Tue 13:00–14:50 N41
  • Timetable of Seminar Groups:
M3130/01: Wed 12:00–13:50 UP2, M. Čadek
M3130/02: Wed 14:00–15:50 UP2, M. Čadek
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2003
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
Mgr. Richard Lastovecki (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Timetable of Seminar Groups
M3130/01: No timetable has been entered into IS. M. Čadek
M3130/02: No timetable has been entered into IS. R. Lastovecki
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2002
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
prof. RNDr. Jan Paseka, CSc. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Prerequisites
M2110 Linear algebra II
Knowledge of basic notion of linear algebra including eigenvalues and eigenvectors.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra is devoted to more advanced topics: the structure of linear endomorphisms (including Jordan canonical form) and multilinear algebra. These topics find applications in ordinary differential equations, differential geometry and physics.
Syllabus
  • Geometry of linear endomorphisms: eigenvalues, root subspaces, nilpotent and cyclic endomorphisms, Jordan canonical form. Polynomial matrices: equivalence, canonical form, connection with characteristic and minimal polynomials and with the Jordan canonical form. Duality: dual vector space, dual basis, dual morphism. Multilinear algebra: tensor product, equivalence of different definitions, change of coordinates, functor Hom and its reletion to tensor product.
Literature
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2011 - acreditation

The information about the term Autumn 2011 - acreditation is not made public

Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2010 - only for the accreditation
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M2110 Linear algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.

M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2007 - for the purpose of the accreditation
Extent and Intensity
2/2. 4 credit(s) (fasci plus compl plus > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. RNDr. Martin Čadek, CSc. (lecturer)
doc. Mgr. Jaroslav Hrdina, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Čadek, CSc.
Prerequisites
M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form. These topics find applications in differential geometry and ordinary differential equations.
Syllabus
  • Afinne and projective spaces: definitions, subspaces, endomorphisms, complexification. Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification. Metric properties of quadrics: main directions, main planes, metric classification. Multilinear algebra: dual space, tensor product, external and symmetric products,coordinates of tensors, functor Hom a its relation to tensor product. Polynomial matrices: equivalence, kanonical forms, connection with characteristic and minimal polynomial, Jordanovým canonical form.
Literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
Assessment methods (in Czech)
Výuka: přednáška a klasická cvičení. Zkouška: písemná a ústní. Ke zkoušce je nutný zápočet ze cvičení.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~cadek
The course is also listed under the following terms Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (recent)