Angličtina pro matematiky IV
COURSE MATERIALS AND HOMEWORK WEEK II.
These are the materials from week II.
Listening 2
http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture10.htm
Answer the Qs.
1. What are the two important things to do?
2. In what did the lecturer´ s mistake consist?
3. Who reminded him of that mistake?
4. Can the columns be independent? Why yes/no?
5. Which conclusion will this lecture reach?
6. How many subspaces have already been discussed?
Translate:
1) Jednodušší příklad jsem nemohl zvolit.
____________________________________
2) Měl jsem si toho všimnout dřív.
_____________________________________
3) Takto můžeme shrnout a opravit minulou přednášku.
__________________________________________
4) Nemusela ztrácet čas hledáním vhodné kombinace.
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Transcript - Lecture 10
Listen to the recording again and try to match sentences and gaps.
a) And that makes the columns dependent.
b) So in my innocence, I put in three three eight.
c) I couldn't have taken a simpler example than R^3.
d) I'll have two equal rows and the row space will be only two dimensional.
e) Wait a minute.
f) And why did she say that?
OK, here is lecture ten in linear algebra.
Two important things to do in this lecture.
One is to correct an error from lecture nine.
So the blackboard with that awful error is still with us.
And the second, the big thing to do is to tell you about the four subspaces that come with a matrix.
We've seen two subspaces, the column space and the null space. There's two to go. OK. First of all, and this is a great way to recap and correct the previous lecture -- so you remember I was just doing R^3. 1) ______________________________________
And I wrote down the standard basis.
That's the standard basis. The basis - the obvious basis for the whole three dimensional space.
And then I wanted to make the point that there was nothing special, nothing about that basis that another basis couldn't have. It could have linear independence, it could span a space. There's lots of other bases. So I started with these vectors, one one two and two two five, and those were independent. And then I said three three seven wouldn't do, because three three seven is the sum of those. 2) ______________________ I figured probably if three three seven is on the plane, is -- which I know, it's in the plane with these two, then probably three three eight sticks a little bit out of the plane and it's independent and it gives a basis. But after class, to my sorrow, a student tells me, "Wait a minute, that ba- that third vector, three three eight, is not independent." 3) ________ She didn't actually take the time, didn't have to, to find w- w- what combination of this one and this one gives three three eight.
She did something else. In other words, she looked ahead, because she said, 4) _________ if I look at that matrix, it's not invertible. That third column can't be independent of the first two, because when I look at that matrix, it's got two identical rows.
I have a square matrix. Its rows are obviously dependent. 5) _____________ So there's my error.
When I look at the matrix A that has those three columns, those three columns can't be independent because that matrix is not invertible because it's got two equal rows.
And today's lecture will reach the conclusion, the great conclusion, that connects the column space with the row space. So those are -- the row space is now going to be another one of my fundamental subspaces.The row space of this matrix, or of this one -- well, the row space of this one is OK, but the row space of this one, I'm looking at the rows of the matrix -- oh, anyway,6) _____________________ The rank of the matrix with these columns will only be two. So only two of those columns, columns can be independent too. The rows tell me something about the columns, in other words, something that I should have noticed and I didn't.
REAL NUMBERS
(materials adapted from Mathematics, encyclopedia articles on http//en.citizendium.org)
1. Have a look at the picture. How do you understand it? Try to explain. How are the words “rational” and “real” used in Maths?
2. Read the first part of the text and explain why the term “real number” was introduced.
3. What are imaginary numbers?
4. What is “a misnomer”?
In mathematics, real numbers are thought of informally as quantities identified with points on an infinitely long gapless straight line. The number zero is one such point; positive numbers are to its right and negative numbers to its left. The integers (positive or negative "whole numbers") ..., −3, −2, −1, 0, 1, 2, 3, ... are real numbers, and points on the line between integers (e.g. 2.75) are also real numbers. The reason such numbers are called real numbers is that mathematicians have found it useful to introduce what are called imaginary numbers—a misnomer—such as , sometimes represented as . The term real number was introduced to distinguish real numbers from complex numbers (so-called because they can be expressed as the sum of a real number and a purely imaginary number, i.e., the product of a real number and i ).
Read two paragraphs below and answer the following questions.
1. There are listed some properties of real numbers. Try to find the properties matching these definitions.
a. ................a real number that cannot be expressed as a fraction m/n in which m and n are integers.
b)...................a real number that is not a root of any polynominal whose coefficients are integers.
c) ...............a real number that is a root of a polynominal equation with rational coefficients.
d) ............a number that is either an integer or can be written as a quotient of two integers.
2. Explain the meaning of two other properties – and ordered set and having the least upper bound property.
3. What does the symbol R represent? What is Rnand R3?
4. What can be real in Maths? Give examples.
Basic properties
A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.
More formally, the field of real numbers has the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the field of real numbers up to isomorphism, and thus allow its other properties to be deduced.
Uses
Mathematicians use the symbol R (or alternatively, ) to represent the set of all real numbers. The notation Rn refers to an n-dimensional space of real numbers; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space.
In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.
Definition
5. One of the important properties of the real numbers is their completeness.
What do you think this means?
6. Read the text and supply the missing information:
a) The reals were introduced because .........................................
b) A Cauchy sequence is a sequence where.................................
c) A sequence is convergent if......................................................
d) The opposite of “convergent “is”.............................................(this is not in the text).
Completeness
The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete. This means the following:
A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements xn eventually come and remain arbitrarily close to each other.
A sequence (xn) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x.
It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: Every Cauchy sequence of real numbers is convergent. That is, the reals are complete.
7. Now study the following examples and decide whether the sequences are Cauchy sequences or not. Explain why you think so.
a) 1, 1, 2, 3, 5, 8, 13, 21, ... . c) x, (x+a), (x+2a), (x+3a)………..
- b) d) 2,4,8,16,32…………. d) 1, 1/2, 1/3, 1/4, 1/5...................