Cramer 's Rule

PropositionLet A = ( at

. .-

. .

ah ) E Du" " "

( here ate IR"

is the

i - the column of A ) be a regular matrix.

Then there

is a unique solution X of the system Ax=b,

AbeDu"

whose components C a , Xu - in ) are given by

anout ( a

'

,a } - → a

" '

,b

, AY'- - -

at ),=

det Ahe

jProof A. X = I '

xjaa b

=/j.

=D dit ( a'

, -

,

b, - ,

a

" ) = out ( a'

, -

,Eh

,x.ad ,

an)j 3

-

P J

fgth column

= ¥,

xjdit la !

-

nai, - at )

linearity of out j p"

with respect to rows

"

Kh column

(= xn . out ( a

"

,- ,

at, -

,

at ) = Xa -olet A

if j t k the matrix has

the km column equal to

another column of thematrix A ⇒ its determinant

Vanishes .

DK

Linear transformation

Let V,

W be two vector spaces and consider an

application T : V - o W.

This is said to be a linear transformation ⇐ b

Fa, per and un e V

Tcxutpv ) = a Tcu ) tp

Tcu ).

The easiest example of such an application is :

n in bn x w

V = Re,

W =D,

At

112N M

T : IR -0 112in

he

Rax1-0 Tox ) = A. x e R

Ff V and W are finite

dimensionalvectorspaces ,

everylinear transformation is represented by

a matrix according to the following :

1. tix a basis in V, Billy, -in} and a basis in W

,BuffWn,

-iwm}

2. for each element VJE Br write Taj ) EW as a linercombination of the

elementsin Bw i

hr

Tcvj ) = I '

avg. wi t je Le ,

2, -

,w }

is 1

The matrix A = ( aij ) represents Tw .net.

the

basis Bv and Bw .

Note : if you change the basis Bv , Bw the matrix

A changes !

Proposition : Let T : V - 0 W be the application above.

It holds

u =jEaxjvj ,

w Tiwi

w .- ten ⇒ I

f ;) . A .

th x he

Note that AE 112,

that is

# columns of A = n = olim V

# rows of A = on = olim Wa linearity of T

Proof : Tcu ) = T ( jaxjvj ) = xj Taj )

outlining = FiEIaijwi.IE?aijxj)wi" )

matrix A- ( aij )m

on the other hand To ) = W =L y ; wi 62 )i=1

The equality of a ) and Cz ) gives that

yi = Ea,

aijxj tie 21.4 - im ) ME

Example : Rotation by an angle D in 1122w

. r .t . Stander

basis Lee , ez )T .

.

Ri - o IR'

T Cen ) = cos O en + singe ,

Toei ) = ( Iff )Tca ) = - since er tender

T ( ez ) = (- find

)^ cos O

Tcez )ez

←g,→

T Cen )cos O - since

.' o

e.

° A = ( since wso ).

What happens if we change basis ?

Consider the following example :

T : 1122- o IR

'

is represented , in.

r.

t-

the canonical

basis of1122 and IR

'

as

ten - ( ! ! ) ( II) .

This means in particular that the image of the vector

e,

= f to ) is T Cea) =/}) = A . eat 2. eats ez

A

goT Teen )

^ Ien>

s

eze

,

<

We now look for the representation of T w . r .

t the followingbasis :

pi

I= f! ) ,

I-

-

fi)

winsIR

'

wi=/! ) ,

Iz=/I) ,

5=1!)

Lwiiwuw's

The image ofviisThi) =/ ! Is) - (f) = f I )

and (F) = - 2WT-262+9WJ- o f ÷)The image of IisTWI= (} Is) (

- f) =/ } )and ( § ) = o .WTto .WIT3-Ismo (§ )

w N ~ ~ ~

With respect to the new basis 14 , vz } and Lwr , waw , } we

hone me representation

B =

fig§ )

Observe now that the vector en of the standard basis

of1122

can be written in the new Luis } basis as

Halil .

HYe

.= I

Tn- I

The( to ) in the basis fee, ez ) is ( "⇒ in the basis C

Te,

T f to ) = ( } ) in the basis ( er , ez, ez )

T ( I:) = § ) ( = ( I ) in the basisErin,%)Now observe that -twi-two-13Wj= (§)÷:÷.

goTce

, ).

~ A •

Vz N U

r o Vr • E353

> n

~ ~Wr s

Ily - he ) v eze

,

Jr,

↳construction of

lntzezt 3 ez

Remark :

Let T : V - s W,

S : W -0 Z.

Fix Bu,

Bw and Bz to

-

be basis in U,

W,

Z, respectively ,

and assume that

T is represented by the mobux A and S by the matrix B,

that is

✓ W W

#Z

I . ÷,the application

SotV -0 Z is represented by

the matrix B . A.

2- = By = B CA x ) = ( BA ) -x.

Change of basis

Let T : V - ow a linear map .

let By = LY ,- Nu }

and Bw =L we,

- ,Wm } be two basis in V and W

,

respectively .

With respect to Bv and Bw ,the inept

is represented by the matrix A e Dum"

?

Problem : Which matrix represents T if we changebasis in V and W ?

suppose to tix BT= LVT, -

,The } andBI = LWT

,- ing

as new basis in V and W, respectively .

We define two metrics of change of basis to be

Se IR" "

and Rx 112mmgiven by

he

S : C Sjw ) I = E Sjw Vj and

j = i

m

Ri ( rie ) WI = Erie wi

i = e

Note thot by ohfiuihiou : the matrix S has as

column he the coefficients of the linear combination

expressing the k - the element of the new basis in the

old one .A hologram sty the matrix R

.

It is possible to prove thot if S express the

change of basis from Bv to Bv,

the matrix

S- t

expressthe charge of basis from ⑤

✓to Bv .

T

✓ -W

ABv - Bw

s T T R

~ B ~

Bv → Bw

T✓ -

amW

A- I

Bu → B B = R A S CA)A

W

s Tads E'FR

~ B ~

B -

•B

✓ W

The matrix B,

which represents T in the new basis

Bv and Bw is obtained from the matrix A representingT in the old basis Bv and Bw

,thanks to CA )

Example ( few pages above )

Will respect to the standard boss 's B = Zen , ez } ,B = Len , ez , 93

1122 1123

Tex ) = (} f If) = As x.

Fix the new basisBag LVT,To }

,

BE = LEE ,wj3,

where E- ( I ),

E- I I ),

in -- I ! )

,ii. I :o)

,

is =/ !)The matrix S i Tn = ten t lez

,

T,

= - ten t l

easy; ; )

The matrix R i

✓a

= t er to ezt Oeg , Tze l eat lez to ez,

wig = tent heat t ez

at : : :)We can compute Ri

'

using Gen p- Zordon

i : : :L : : " % : :L ::÷i( E !? I ! I so not to :c !!÷ ) .

Finally the nohix B =R'

t

A S =

in :÷÷H:¥HiiH÷ :).

OrthogonalityConsider a vector space

V endowed with a

scaler product C e

,- ) and denote by Hull = the

induced norm.

Orthogonal systemA system of vectors I we , - ,

Win } , Wj to tji said

( i ) Orthogonal system ⇒ C Wj ,Wu ) = o j 't o

( the elements of the basis are orthogonal )to each other

Cii ) Orthonormal system ⇒ ( Wj , Ww ) = Sjw( besides orthogonality one requires also that

11 wjll =L t j )

Ciii ) Orthonormal basis ⇒ they are an orthonormal

system and they form a

basis of V

Every orthogonal system is mode of linear independentVectors : let an ,

- - -

, dm EIR be such thot

o =g€ndjwj . Taking the scalar product of thot with Wu

o= I Wu ,o ) = ( wa

, jedjwj) = GII,

dj ( Wu, Wj) = dull Wall

'

-0 g- O.

whichproves the linear independence .

Note that the representation of a vector with respect to

an orthonormal basis is :

m

v = I Cr , Wj ) Wjj = a

m

In fact , given v= I ajwj , toking the scalar product

with Ww we have J = '

( u, wa ) = ( ¥,

ajwj ,Wu ) = It

,

- j l wjiwn ) = ,¥aj%= an

Gram - Schmidt Orthogonal talion

Given a system IT,- ,Nm } of linear independent

vectors in V,

we want to construct an orthonormal

system Lun , - , um } such that

spew Lori ,- arm } = Spen Lui ,

-

, um } .

Gram - Schmidt recursive method :

Step

@=D-

my = IsHrh It

Step@

to lets )

like' Yeti - fi,

( duty Mj ) Mj and una = line" unwell

.

Note thot rim,

t w . t .

E 21 , - ik }

( iii.in ;)= ( run !u - ( Eactuary )uj ,ni ) -

= ( Nuu ,hi ) - ( rate ,

hi ) Huitt = O.

÷

since Uma = Truth,

we also have wht ,Lui tie It , - ik}

.

Kliment