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Lecture 1

MAT247H1 Lecture Notes - Lecture 1: Linear Map, Spanx

Department
Mathematics
Course Code
MAT247H1
Professor
Fiona T Rahman
Lecture
1

Page:
of 2
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:VāVbe 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 āR}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=ā2
hx, xi= 4 hx, x2i=ā2hx2, x2i= 3
Let W= Span{x2, x }.
b) Let V=C4with the standard inner product. Let
W={x= (x1, x2, x3, x4)āV|ā2x1āx3= 0, x1āix2+x4= 0 }.
c) Let nā„2, V=MnĆn(R), with the inner product hA, Bi= trace(ABā), A,
BāV, and W={AāV|A=Aā}.
2. Let W1and W2be ļ¬nite-dimensional subspaces of an inner product space V. Assume
W=W1+W2={vāV|v=w1+w2for some w1āW1and w2āW2}.
a) Prove that if Ī²is a basis for W1and Ī³is a basis for W2, then Ī²āŖĪ³is 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 W1āWā„
2and T1T2(v) = T2T1(v) = 0for all vāV.
c) Suppose that W1āWā„
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)|xāW}and U(Wā„) = {U(y)|yāWā„}.
Suppose that Uis invertible and U(W)ā„=U(Wā„). Prove that UT Uā1is 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)āVsuch that hf , gi=f0(0) āf(1) for all f(x)āV.
5. Let V=MnĆn(C) with inner product hA, Bi= trace(ABā), A,BāV. Fix CāV.
Deļ¬ne a linear operator TC:VāVby TC(A) = CA.
a) Find the adjoint (TC)āof TC
b) In the case n= 3, ļ¬nd all CāVthat have the property that (TC)ā=āiTC.
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6. Suppose that Ī²={x1, . . . , xn}is an orthonormal basis for an inner product space V.
Let T:VāVbe a linear operator. Deļ¬ne a function U:VāVby
U(x) =
n
X
j=1hx, T (xj)ixj, x āV.
a) Prove that Uis linear.
b) Prove that U=Tā.
7. Let Vbe a ļ¬nite-dimensional inner product space. For each x,yāV, 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 āC}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:VāV
be the linear operator on Vdeļ¬ned by T(a+b x) = āa+b+ (ai āa+b)x,a,bāC.
a) Find Tā(a+b x) for all a,bāC.
b) For each cāC, let Uc=T+cT ā. Find all complex numbers csuch that
9. Let T:VāVbe a linear operator on a real inner product space V.
a) Let U=TāTā. Prove that Uā=āUand hU(x), xi= 0 for every vector xāV.
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:VāVsuch that
hU(x), xi= 0 for every vector xāV.
10. Suppose that f(t) = antn+Ā·Ā·Ā·+a1t+a0, where a0, . . . , anāFand 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 Tāis invertible and (Tā)ā1= (Tā1)ā.
b) Suppose that Tā=cT ā1for some scalar cāF. 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 ā1, 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 ā1if and only if for every
orthonormal basis {x1, . . . , xn}for V, the set {ācā1T(x1), . . . , ācā1T(xn)}is
also an orthonormal basis for V. (Here, ācdenotes the positive square root of
c.)
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