MAT 247S - Problem Set 2

Due Thursday January 29th

NOTE: Questions 1a), 2b), 3, 8 and 12e) will be marked.

1. For each of the following inner product spaces V, let T:VāVbe the orthogonal

projection of Von the subspace W. Compute T(v) for all vectors vin V.

a) Let V=P2(R) = {f(x) = ax2+bx +c|a, b, c āR}be the vector space of

polynomials of degree at most two, having real coeļ¬cients. Suppose that hĀ·,Ā·i is

an inner product on Vthat satisļ¬es

h1,1i= 2 h1, xi= 2 h1, x2i=ā2

hx, xi= 4 hx, x2i=ā2hx2, x2i= 3

Let W= Span{x2, x }.

b) Let V=C4with the standard inner product. Let

W={x= (x1, x2, x3, x4)āV|ā2x1āx3= 0, x1āix2+x4= 0 }.

c) Let nā„2, V=MnĆn(R), with the inner product hA, Bi= trace(ABā), A,

BāV, and W={AāV|A=Aā}.

2. Let W1and W2be ļ¬nite-dimensional subspaces of an inner product space V. Assume

that W1ā©W2={0}. Let

W=W1+W2={vāV|v=w1+w2for some w1āW1and w2āW2}.

a) Prove that if Ī²is a basis for W1and Ī³is a basis for W2, then Ī²āŖĪ³is a basis for

W.

For parts b) and c), let Tjbe orthogonal projection of Von Wj,j= 1, 2, and let U

be orthogonal projection of Von W. Let T=T1+T2.

b) Prove that if T=U, then W1āWā„

2and T1T2(v) = T2T1(v) = 0for all vāV.

c) Suppose that W1āWā„

2. Prove that T=U.

3. Let Wbe a nonzero subspace of a ļ¬nite-dimensional inner product space V. Let

Tā L(V) be the orthogonal projection of Von W. Suppose that Uā L(V). Let

U(W) = {U(x)|xāW}and U(Wā„) = {U(y)|yāWā„}.

Suppose that Uis invertible and U(W)ā„=U(Wā„). Prove that UT Uā1is the or-

thogonal projection of Von the subspace U(W).

4. Let V=P2(R), with the same inner product as in question 1a). Find the polynomial

g(x)āVsuch that hf , gi=f0(0) āf(1) for all f(x)āV.

5. Let V=MnĆn(C) with inner product hA, Bi= trace(ABā), A,BāV. Fix CāV.

Deļ¬ne a linear operator TC:VāVby TC(A) = CA.

a) Find the adjoint (TC)āof TC

b) In the case n= 3, ļ¬nd all CāVthat have the property that (TC)ā=āiTC.

1

6. Suppose that Ī²={x1, . . . , xn}is an orthonormal basis for an inner product space V.

Let T:VāVbe a linear operator. Deļ¬ne a function U:VāVby

U(x) =

n

X

j=1hx, T (xj)ixj, x āV.

a) Prove that Uis linear.

b) Prove that U=Tā.

7. Let Vbe a ļ¬nite-dimensional inner product space. For each x,yāV, let Tx,y be the

linear operator on Vdeļ¬ned by Tx,y(z) = hz , yix. Show that

(i) Tā

x,y =Ty,x.

(ii) trace(Tx,y) = hx , yi.

(iii) Tx,yTu,v =Tx,hy , uiv.

(iv) Under what conditions is Tx,y self-adjoint? Explain your answer fully.

8. Let V=P1(C) = {a+b x |a, b āC}be the complex vector space of polynomials in

the variable x, of degree at most 1, with complex coeļ¬cients. Suppose that hĀ·,Ā·i is an

inner product on Vand Ī²={x, x + 1 }is an orthonormal basis for V. Let T:VāV

be the linear operator on Vdeļ¬ned by T(a+b x) = āa+b+ (ai āa+b)x,a,bāC.

a) Find Tā(a+b x) for all a,bāC.

b) For each cāC, let Uc=T+cT ā. Find all complex numbers csuch that

Uc= (Uc)ā. (Please explain your answer fully.)

9. Let T:VāVbe a linear operator on a real inner product space V.

a) Let U=TāTā. Prove that Uā=āUand hU(x), xi= 0 for every vector xāV.

b) Suppose that Vis ļ¬nite-dimensional and has dimension at least 2. Using part

a), prove that there exists a nonzero linear operator U:VāVsuch that

hU(x), xi= 0 for every vector xāV.

10. Suppose that f(t) = antn+Ā·Ā·Ā·+a1t+a0, where a0, . . . , anāFand F=Ror F=C.

For Tā L(V), deļ¬ne f(T) = anTn+Ā·Ā·Ā·+a1T+a0IVand ĀÆ

f(T) = ĀÆanTn+Ā·Ā·Ā·+ ĀÆa1T+

ĀÆa0IV. Assume that Vis a ļ¬nite-dimensional inner product space. Let Tā L(V) and

let U=f(T). Prove that Uā=ĀÆ

f(Tā).

11. Ā§6.3, #13.

12. Let Tbe an invertible linear operator on a ļ¬nite-dimensional inner product space V.

a) Prove that Tāis invertible and (Tā)ā1= (Tā1)ā.

b) Suppose that Tā=cT ā1for some scalar cāF. Show that Tis normal.

c) Show that if Tis as in part b), then Ī»ĀÆ

Ī»=cfor every eigenvalue Ī»of T. Explain

why this implies that cis a positive real number when F=C, and also when

F=Rand the characteristic polynomial of Thas at least one real root.

d) If F=Rand Tā=cT ā1, then it is possible that Thas no real eigenvalues. In

this case, how would you show that cis positive?

e) Let cbe a positive real number. Prove that Tā=cT ā1if and only if for every

orthonormal basis {x1, . . . , xn}for V, the set {ācā1T(x1), . . . , ācā1T(xn)}is

also an orthonormal basis for V. (Here, ācdenotes the positive square root of

c.)

2

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###### Document Summary

Note: questions 1a), 2b), 3, 8 and 12e) will be marked: for each of the following inner product spaces v , let t : v v be the orthogonal projection of v on the subspace w . Suppose that h , i is an inner product on v that satis es h1, 1i = 2 h1, xi = 2 hx, xi = 4 hx, x2i = 2 h1, x2i = 2 hx2, x2i = 3. Let w = span{ x2, x}: let v = c4 with the standard inner product. W = { x = (x1, x2, x3, x4) v | 2x1 x3 = 0, x1 ix2 + x4 = 0}: let n 2, v = mn n(r), with the inner product ha, bi = trace(ab ), a, B v , and w = { a v | a = a }: let w1 and w2 be nite-dimensional subspaces of an inner product space v .

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