16 Feb§ 1.9 Matrices of linear transformations.

Recall that anmxn matrix A is determined

by where it sends In,

Iz, ... ,

En,

and these vectors

Aei , . .

,,

Aeon are simply the columns of t ,this

gives a powerful tool for finding the matrix

of a given linear transformation : figure out

where Ee," yer go and

arrange the

resulting column vectors into a matrix !

Example Find the matrix of the rotation 90° ccw

about the origin .

soution do ] maps to (9) : ^¥[ 9] crops to [ to] : ¥f

some matrix is [9 to] .

Done !

Example Find the angle - O rotation matrix.

←

t.EE#f*hEotaFaeIE:..wfa

so we set CohoInooo].

twitter.TEg.ee?mImTxFnam3gaa.rotation matrix and using it to multiply each

ftp.] for in, ...

,n

.

n m

Definitions We say T : R → R is sunjectieor onto if range CT ) = Rm,

i.e. the columns of Tspan R?

And T is wijective ,or one-to-one

,if tcxktcg ) ⇒ I=j .

"

T##¥±±⇒#f#H##¥¥tII¥o

⇐÷¥IE#*####E¥¥##FF#n¥#ft

theA linear transformation TR

"

→RM

is injective if and only if T (E) = 8 has

a unique solution .

thepiratebay.ms?EoghsyW#swtWy?ifTCxT=0=tCy

) for Ity ,then

already that's two distinct inputs givingtie same output , so T is not injective .

Conversely , if TCE ) = TIJ ) for some Ity ,then

T (e) - T (g) = 0 iamb"

TH -5 ) = 8 and so I-5 is

a nontrivial answer to TC ? ) = J.

[ Recall thata sweat

"

A implies B"

Is logically

equivalentto its

Contra positive"

not B

impliesnot A

.

"

Sp showing noimjectioityimplies nontrivial solutions tells as that no nontrivial solutions

impliesiujectioity!

)exercise show that TC x. g) = ( xty ,

× - y ,Zx )

(nidea)

Is injective . solution: snow ( ft ) has a pivot in everycolumn ,

A Exercise If T : R"

→ Rm is injective and surjective ,

°°then N=M . ( Idea : count pivots )

§ 2.1

Matiixopaatiouswedefine matrix addition and scalar multiplication

like for vectors :

gave to be the same size

adder ( 4253 ) + ( stats) = ( j} } )scald

mean 31'

I:) . this:bThese operations do what you think :

ATB = BTA

; r ( s A) = Crs ) A

At&tC) = HeB) tc ; r( At B) = r At RB.

You can prove any of these if you really want by

just writingit art with symbols like A = Paid÷ia9m!D,

and using corresponding properties of real numbers.

What's more interesting Is multiplying two matrices .

Consider that if we want to apply a matrix B to

I and then multiply the result by A, we 'd naturally write

Ifbudgeting

ABI It would be cool if we defined@ then multiplythat by A

AB so that this expression can be calculated

either way: A ( Bx ) WE

'9t(A B) x

.In function

language ,we want the matrix product

to correspond to the composition of the !,9aa2yM

corresponding linear transformations .

so consider : AnalyserACBI ) = A ( X ,B,

+ . . . + xn⇒fbieaity

= x, AB,

+ . -. t xn Aton

deft of matrix . (vector productin reverse

±[ AD,

..

.. Aton] I

The upshot : matrix - matrix product works just like matrix - vector

product , one column of B at a time :

Solution:

example find KEEFE.IM :[Ata AM AHHH

= toss 448 % :]

multiply entriesWe can simplify this somewhat : in pairs

jg§qjgE{

"

To find the Cisjlth entry of AB,dtotesmthe ith row of A with the jth column

I"

{ of B

It column

innitA ] [ B ] = (NAB ](i

,Marty

n←

these have to be equal toycomposition to make sense.

n M

Note : If B : RP → R and A : R → R, then

the composition goes from RP toRm; likewise :

( mxn ) . ( n xp ) = mxp⇒"→ IRM o RP → Rn RP → Rm

theidentity matrix

.

We define In to be the n n matrix with

f 's down the main diagonal & Zeroselsewhere= Fo:] , ±=[ to:O:D

.tn#HLE?nd?oEw

the matrix . matrix product also'

does what you think'

LtB) C = ACBC ), (A)B = r(AB) ,

etc.

BUT ! ! !ExesShow that matrix multiplication Isvotedcommutative .

Solutionlet'schoose some numbers

and go at it : [ ' ⇒ [ 5,8] = Classy

[ Egg = Eg 3yd

Solution ( geometric )

rotate 90cm reflectcomposition |- ~ |-Fiefx - axisEighmey:*

× . axis

-

A"

= AAA . . . A. cfnly

works if ADefinition-

is square )ntimes

Definition The transpose of A, denoted

AT,

is obtained by forming a matrix

whose rows are the columns of A,

in the same order.

example [ '

y ;}I=[ts¥].

2×3 - 3×2

Tadd ( At B) T= Att BT ( easy )

CAT )T = A easy)

→ DoD (AB)T = BTAT ( you can figureIt out )

Transpose distributes & reverses order-

a