Today : § 2.1 : Matrix OperationsNext : Midterm 1 Review

Reminders :

My Math Lab Homework # 3 : Due TONIGHT.

Midterm 1 : THIS Wed,

Jan 31,

8- 10pm .

covers § 1.1 - 1.5,

I. 7- 1.9

practice midterms posted on webpage

seat assignment posted on Triton Ed.

'

§ 2.1 : Matrix operations

We can treat

matrices1 of the same size ! ) like

vectors,

and add them and multiply them byscalars

.

Eg.

A .

. kill Bit ::D off;)2A . B = total - few . tinted

→kI'#Bt 2C NOT DEFINED !

Addition and scalar multiplication of matrices

behave nicely together ,just like for real numbers :

At B = B + A rl At B) =rA+rB( At B) + C + A + ( Btc ) ( rts ) A =

✓ At SA

At 0 = A r ( SA ) = eg A

±no matrix

of appropriate Warring:

dimensions( D AH only when sizes match !

(2) Beware : noteoey property

from real algebra works formaths

..

.

Matrix addition and scalar multiplicationcorrespond to addition and scalar multiplicationof linear transformations

'

Is :# → 1122

Eg. A . .by?lB=l92z 's)

z

Tlx.

) = AI Six )= By I £ R

( Tts ) b) = TCIHSCI ) =A§tBx=(A* B) I

(rT) b) = r.TK.

)=rA # = @A) I

Matrix MultiplicationIf T : R

"→ Rm and S

:Rm→Rkwe can compose them

( sot )c⇒Rn

TRm

SRk = S( Te ) )

÷Sat :R

"

IN.

Notice :

± g-c- Rm

Sotlutv )=S(Tly+±))=S(

th,)+Tk) )

=SC It .y)=SlDtSly)Sotlrv ) = ratty =S(TluD+S( TKD

set respects addition & scalar multiplication =§oT)u ) HSOTJ b)2 a linear transformation !

T :Rn→lRm S

:Rm→|RkTh ) '- Ax

.

Sly )=By" '

Mid tm n k MA=1ga...

. an ]

S.TK. )=S( TH ))=S(

AI)=S(

xigitx ,G+-→×ngD

=x,S( g.) tkslqdt .. . txnslgd

=x, Bg ,txzBGt - " txnbgn

Btkstandadmatnxof jµ9BG"

.BG#yIT

.

Definition : If A is mxn and B is kxm

the product BA i3 defined by

BA = B[ Qi G - - . Qnl := ( BQ , B.az - Been] .

⇐Might#YBH Bet .tt#tIYB:YHEttM

3×2 2×2= | fg LY ) = BA

A 13

E "

( t3jf4j2g )AB does not even make

2×2×3×2sense .

Properties of Matrix Multiplication

AIBC ) = LAB)C

ALBTC ) = ABTAC

CBTGA = BATCA

( r A) B = r ( AB ) = A ( RB )

= A.tn#deFthgmatrix

If A is mxn,

Im A = AMATHBEYMD )

mxm mxh Mxn nxh

AB ¥ BA in general .

Even squarematrices don't generally commute

.

A =

147] B =/

29)

3 4 83

AB :

l ;Etzel 4638331

B As1 }%lYIl =p ?:)

TransposeT

Hyam:&:b

mxn

nxm

' ⇒ Eton:tI÷g;H¥FHIM6 -1

symmetric.

PropertiesIf A is mxh

,then AT is nxm

(At)t = A HtB)T= At + Bt

( ABI = Bt At (rAJz RAT