Week 4 : Affile OEoM&rzr
&cJs  iL   be  a   vector space  &ve\r IK - JR o<l (ľ
a*    affine   subQpace of  ic f  if   it is   of &ke form
/frls =      1ľ - í V-t Ar etut      e frj Where,   T e it   astd V Q. ti   cs a vector scibspac^
Remark   £le lhot rathe\r tham vectors.
äff Cm   cub spaces   ť*   R a*e-fji)   all    points.        lhhh\so4$.e   {ľ = U?}
(í)    l/'Hes.     Ih   ^ ^ í* * //W 1
(D
A   is  4 to&ért*    k* n,    be?1   such éh<ib
i i
X ( A) =   h C All*) (h.  s&hds for \ra*t)
\h this,  case    the.   Oc^stem of-   /ľne*r ďfUféibtis A*- b      tas   4.   Solution    ahčt Oils is ftoérenpéý Let q e Tg*  be   <Z eetutiw  e>+ the e%*t*»> fy y Ä b . Them
(TMôtJtvn :    Anc^   solution X   of  A**b is of the f-oŕtn where.      is <z solu-
tion    of Ažr-O.^
lvi lbe definition    sty, ^ T+V y/ie
T is noů áteéertoľneot   otnifuelß.  5ee tř*e-
®
Lemj^    The   svbs^ce   V   (h   kk* atefrWh*
Alfjjie    Combi ftdbioM    £>f poCvihs sptfce* iL.    Then    the uffih^ comhCMhCo\A
(4-b)       6-3 = A + y CB-A) . 4// affine   ce>i*biwibu!>n£>   fwm a lt'»e*
tCA-B)
A        l *
AofAji*-. An     fs     £he po{\t
É k A,
lbe   h4 ve.
2j ér - ±
i=0
t =0
-f .... -t A>)
Theorem 4      \n eveir    a^itie euhepace-Wh  wľbh   any   jpoínbs   4«, Ai,& tHs
MS
/Us
^hcA{e^     for =
i-0
1^0
ľrg?č?f-:    ^ - V* V   /u***. 1ľ £>4t C* ^ sector
(-0 i"=0 ť°0 l~0
ist?
Theorem 2, Lei 41% be a honesty subset o£   tke  \r&cbo\r space %   šuch   thai, far
Vra>o^_    The<re C&       poiní Pe- OU^ Define-ft  success,    ho    chow  M   & es * wto*
Leb     A-T, * V, At>4e<re A,B,Te^.
Then
(a-t) t (t-v) - fe(iA* žzyfpj. Tíé^
g syn,
^ =    -ja. At _ -p é ^
v--- v (2j
Sč> ife cah   q(ve   &n equivalent def-in*'hov) of   d f-plne,   ßuhspzce* OŤl    4£>   <z nwenjobtj
suh&eb of the vecbov gpace 40 such bh&b foir eue^ A and 3 in /m/ A£3/ also I foe,  AS  //« ^
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¥a\ra HebHc    description    o£ gff/k<g subspict.
/yyi  ^ T t V       V  has  i  basis vu y u
£o ev<?^    p01 h-h   M er da h   be i&trcérůen
in bn e for vn
M » Tt -if v, t éi. ľ, + ■■ ■ ■ 4 (T) Jhira toetH'c   olesarípkíovt  a f a. Uns ťn ^ (I) Va^am&bnc.  des en pi m  of- 4- P^ihe ľn ^
SÝTI = I %é IK^ ( A*=yj
(Z)    A /i»e in
<?2-< ^ * * 2,1,*2, *?*3 *3   * k»2L
l^ - z    c4*, ****** \
fs   eas^     /i /s sufficiertt   bo  solíte. 6~4e-
Sappoše-       A y =■ b <t holubi on
= (2,3,, f, i) + H 3,     Oř /) f s (H    -2/ 0)
CtescWpbioiA
/m   CoowliWL ires :
"ft
£F x   = C t * p
as
14nib v*4.bH'i>
4*		
-4	<9	b
(e / c i y> )  ~->
frail h<j tt>\)(/
ivi Such <i wai^   bvxah    £ /s   /«  echelon form
Without row. 1
Then    Y ^fct'&ft'es efuebt'ons.
At * =
Die  $>eoovicl etfu^kton   /s t^e
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Ihb&r£*cét0»    of   qfffhe subjptcez
\v[   exer^ses    ure   m&eh    three, posľbi/ihes;
(i)    (fcrtAj SM b £p£ce<L    (ire  qi\ev\    frq   <z Q^čbe*) of &cfiA<ih[0na
(íL)   OH h#š  4 paravnebric úescn'pbioYi
on \  / ~ T>+é, 4^
^/>eh    kej   4  scfčé&toi of etfuci-bi'onz
T^r   y   /h the >4>^^i)
ľ____
So \x/e ^et   4   éyséem   of efučiéťovs foir  hf/ 64- .
(3) Om a 71 are q(v*v\ b<ý yatemebvt'c.
description :
Vo
m, : Q   ŕ S, k, -r sz
Tkev\
H/e solve  éh& £^£bshi   ín   (4 n in even s.
íii ti, é3/ S,, sí ;
/f   w e  <?<?fc Solubľov
S i = (X.
connection of ůwo affine zpicxsz Hl 4*4 %   t$>   demoted b^x
syn u &i/
av\A Ľ b ts> -the £w# ll^h affine £ubzpic&> containing   foth   tyt  a.wl ty.
^ :   T* f a** /H> ' Q*W~t them 4M u ty, ;   p y CS-'PJ t 4ŕ+ P"
uH<?he    c q-t]   f s       Iľheettr envelope af
(1)   W  C 41    or    4L C
(z) ?«»*lle|     t f     /M a/H ~ (p , bul
2r(0fl) ^žfo)   *r   žíl) £
(s)   Ihbeirsecbtaq   t£     4n A m *0 ( y«b