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Math 416 - E13/F13 Matrices to solve linear systems 01/28/
L-as.tk: system of^ linear^ equations.
: X , , ✗ z , - - - , ✗ m
Eq:^ A^ ,^ , X^ ,^ +^ a^ ,^ ,^ ✗^ <^ +^ -^.^ +^ Aim^ ✗^ m =^ b^ , Az (^) ,^ ✗^ , t^ Azz^ ✗^ < +^ -^ -^ - t^ Azm^ ✗^ m =^ bz : an (^) ,^ ✗^ ,^ +^ Anz^ ✗^ < +^ -^ - +^ Ann^ ✗^ m =^ bn^. where (^) aij , bi are real^ numbers^ for (^1) E is n (^) , (^1) Ej E n (^). Today :^ Develop^ systematic method^ to^ solve^ them^.
To a (^) linear (^) system , we^ associate^ a
coefficient maturin^ :^ A- = (
am an - -^ Ain
Az , Az , -. Azn
)
) : (^) :
Ami am , -^ -^ Amn
b ,
a vector^ of constants^ :^ b^ =
(
b
;)
← Column^ vector
bin
and (^) an augmented matrix (^) ( A (^) / b)
a , , 9,2 -^ - Ain b ,
= (
Az - , (^) - Azz- (^) -.. - Azn- - be- l am^ ,^ am^ ,^ Ann^ b)
Enamf
✗ (^) , -1 (^2) × 2 +^2 × 3 = 4 ✗ (^) , + (^3) ✗ (^) , + (^3) × 3 =^5 2X (^) , +^6 × 2 +^5 × 3 =^6
This system has A- =
( LI^?^3 } 3)
-^ b^ =
( Eg^ )
The augmented matin ( Alb) is
,
a :}^ ;)
Decoding
Given an^ mxn matrix^ A and^ a size mxl
column vector b , we associate a linear system
LS (^) ( A. (^) b).
Also ,^ works^ for augmented matrices^ LS^ ( Alb) .
Enamf •^ LS^ ( ( L? :)) =
× (^) , -13×2--
2 ✗ , + ✗^ z =^0
• hrs ( ( LL ) ,( :)) =
× , -12×2--
2X , =^ I
Definition.^ A^ matrix^ M^ is^ now^ equivalent 6- to (^) N if there is a (^) sequenceotrow operations that^ turns^ M^ into^ N^.
Not:^ Needs^ a proof to^ show^ that^ this^ definition is (^) symmetric in^ M and^ N (^).
theorems If^ M^ is^ now equivalent to^ N^ , then the linear^ system Lslm) and^ LSCN) have the same^ solution^ sets^.
¥ Need^ to^ check^ that^ a single now^ operation does not (^) change the solution set^ of the corn.^ linear^ system.
- Row (^) op 1) is (^) fine (^) , as it (^) corresponds to (^) reordering two (^) eqm.
- Row (^) of 2) just scales the assoc (^). egn.
- (^) For (^) op 3) (^) , it (^) is (^) enough to (^) consider the case^ of^ adding rows^ without any scaling.
( The^ general case^ follows^ by now^ op^ 2)^ )
on the (^) eqn version (^) , we^ have
F- , :^ As^ ✗^ it^
- ' + (^) anxn = { a^ ,'^ ✗^ , +^ - -^ +^ aixn^ =^ b^ '
Ez : {
a ' ✗ it -^ -^ + an ✗ n = b
ca, +^ a^ ,'^ )^ ✗^ , +^ -^ +^ (^ anta;) ✗^ n =^ btb^ '
If numbers^ (^ ×^ , (^) ,^ ✗^ z (^) , - -.^ ✗^ n ) (^) satisfy E, (^) , then (^) they also^ satisfy F-^.. Conversely ,^ if^ they
satisfy E.^ , they also^ satisfy^ E,^ as we (^) can subtract^ the (^) first (^) eqn from the second egn of^ F-z. So (^) , all three^ kinds^ of row (^) ops donot change the^ Sol^ :S.^ set^ of^ the^ corn^.^ linear systems.
an
ReducedRowEchelonForm_ A matrix^ where
- All^ zero rows^ are^ at^ the^ bottom^.
a) The^ leftmost^ entry of^ every non^ - zero now is (^1) ,^ and^ is^ called a leading 1.
- A^ leading 1 is^ the^ only non^ - zero entry in^ its^ column G) (^) Suppose entries^ (ij) and^ (^ sit) are leading 1 's^.^ It^ s^ >^ i^ then^ t^ >j^.
