Discussion 19-

Note: For all v. w EV,V a finite -dimensional

inner product space, we have

<Titus , w> = cw.tt#D=sTLwTv7=-v.TCws7

This fills a gapin a proof from last time.

Trolley: Trueor false?

a) w = { ( Tg) EAT I atlotc =/ } is a subspace of 1133.

b) A subset of alinearly dependent set is linearly dependent.

c) If span Cv , ,... .vn) =

V,then each vector

in V canbe written

Uniquely as alinear combination

of ( v. .

- --Nn) .

d) It x=CY,... .vn)

,spank) =V,

and T : v -7Wis linear, then

span (Thi , . . . ,Then)) = in CT)

e) det : lhnxncF)→ F is linear

f) If dimv-_ n and Tiv→

v has ndistinct eigenvalues,

then T is

T - IR" -7113" be linear.

Then the

diagonal izable .

g) Let B be a basisfor Bin

,

and let

eigenvectors forT are the same as the eigenvectors for

[TIP .

Solution : a) False . e, ,e,EW,

but e. tea #W

.

b) False . a :={ l 'd , 191,1 ! ) } ER' is 1. d .

but { l 'd , CTB is not

c) False .

Take a as in Cbl.

Then C ! ) = I - C '

, ) = I - ( L ) ti - (9)

d) True , proved in notes .

e) False . detfcggltcgo.D-i-to-detftodtdett.ci )f) True . For each eigenvalue X

, I ⇐ gme.IT#jFIitysamn9ftfFI7tys1 ⇒ geom.

mult-alg.mu It

g) False. If r is an e - vector of T,then [Dp is an e -

vector of IT],

Prolate : Let V be a finite - dimensional vector space, let a# REV, andlet

W = { TELL v.v) ITU) -03

a) Show that 9 : Luis→V is linear .

T-Tlv)

b) than that w is a subspace at Lcu ,v)

c) Let n=dimV .Prove that dimw

-_ n'- n

.

d) Prove that he is surjective

e) Let XER ,Xto .

Let Z = {TELCV , V)Itu)⇒ v }

.

Is 2 a

subspace of LCYV) ?

Solution : a) Let 5.TELLYv) and c. c-

IR .

We have

4 Ccs TT) =Ccs TT) (

v) = c. Slv) tTcr) = cels

) t lect)

so 4 is linear .

b) since W= kerbw is a subspace .

c) Since veto,we can extend

Cr) to a basis f= Cv, vz, - → rn)for V.

Since 4 :L CVN)→ Munck) is an isomorphism , dimlw) - dimLYND .

T 1-7 ET]p

We 'll show that Ucw) =Xi={ AElhnxncpill the first column

of A is all zeros }.

Let TELCv.rs .we have that

TEW ⇐ TLV) =o ⇒ [Ths]p=o⇐ [T]p[v3p=o⇒ET2pei=o⇒ETTpEX .

Thus Ucw)=X.

Thus dim (w) = dimlx)=n-- n

.

d) we havena = dim ( L (v.v)) =

dimlkerle) t dim lime) = n'- n t dim lime)

⇒ dim lime)=n ⇒ im4=w

e) No,XIREZ , but 2X1v #

Z.

fraktem: Let A C-Mnxnlf) .

a) suppose XEF is an e -rat of A -

Let k Z l - show that Xk is an e- rat of Ak.

b) Suppose A is nilpotent, meaning Tmz I sit. Am =D.Prove that o is an e - rat

of A and that A has no other e - rats .

c) suppose thatA is nilpotent and Ato.

Prove thatA is net diagonal .

-cable .

Solution : a) Let v be an e -vector ofA with e - ral t . By induction

on k,

we have Aku = Ak- '

( Xv) = X Ak- '

v = X ( Xk-'

v) = Xkv

so Xk is an e - rat of Ak.largest integer

such that Aku to( this exists

so o is an e -rat of A w/ e

-

vector Atv.b) Pick otrev .

Let k be the

because Ak -- o ) . Then A ( Aku) = 0,

suppose Xtois an e-

rat at A w/e -vector v . Then

0=0 -

v =Amv = Dmv to

1.

Thus a is the only e - rat at A .

c) By clot,o is the only e -rat of

A . So A is diagonalizeable ⇐dim CED = n

⇐ Ea = Fn ⇐ Kerl A) = Fh# 17=0

So A is not diagonal iz able .

Problems. Let A =/! I, § ,)

.

Find complex matrices Q and D

S.t. Q is invertible , D is diagonal , and A= QDQ

'

!

Solution-

:

Aah) =det(I I )- I - X

=L - i - x) det I! )

= - d't 1)Htt)

=- ( x - i)CXti)Htt)

af :o) - if:) ⇐ ¥÷÷: (1) =L: ) is a solution

soft)eEisimilarly , ( I ) EE - i and (9) EE ,

→ o=(÷÷ !) a--f! ! !)