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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:VVbe 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 R}be the vector space of
polynomials of degree at most two, having real coeﬃcients. Suppose that ,·i is
an inner product on Vthat satisﬁes
h1,1i= 2 h1, xi= 2 h1, x2i=2
hx, xi= 4 hx, x2i=2hx2, x2i= 3
Let W= Span{x2, x }.
b) Let V=C4with the standard inner product. Let
W={x= (x1, x2, x3, x4)V|2x1x3= 0, x1ix2+x4= 0 }.
c) Let n2, V=Mn×n(R), with the inner product hA, Bi= trace(AB), A,
BV, and W={AV|A=A}.
2. Let W1and W2be ﬁnite-dimensional subspaces of an inner product space V. Assume
that W1W2={0}. Let
W=W1+W2={vV|v=w1+w2for some w1W1and w2W2}.
a) Prove that if βis a basis for W1and γis a basis for W2, then βγis 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 W1W
2and T1T2(v) = T2T1(v) = 0for all vV.
c) Suppose that W1W
2. Prove that T=U.
3. Let Wbe a nonzero subspace of a ﬁnite-dimensional inner product space V. Let
T∈ L(V) be the orthogonal projection of Von W. Suppose that U∈ L(V). Let
U(W) = {U(x)|xW}and U(W) = {U(y)|yW}.
Suppose that Uis invertible and U(W)=U(W). Prove that UT U1is 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)Vsuch that hf , gi=f0(0) f(1) for all f(x)V.
5. Let V=Mn×n(C) with inner product hA, Bi= trace(AB), A,BV. Fix CV.
Deﬁne a linear operator TC:VVby TC(A) = CA.
a) Find the adjoint (TC)of TC
b) In the case n= 3, ﬁnd all CVthat have the property that (TC)=iTC.
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6. Suppose that β={x1, . . . , xn}is an orthonormal basis for an inner product space V.
Let T:VVbe a linear operator. Deﬁne a function U:VVby
U(x) =
n
X
j=1hx, T (xj)ixj, x V.
a) Prove that Uis linear.
b) Prove that U=T.
7. Let Vbe a ﬁnite-dimensional inner product space. For each x,yV, let Tx,y be the
linear operator on Vdeﬁned by Tx,y(z) = hz , yix. Show that
(i) T
x,y =Ty,x.
(ii) trace(Tx,y) = hx , yi.
(iii) Tx,yTu,v =Tx,hy , uiv.
8. Let V=P1(C) = {a+b x |a, b C}be the complex vector space of polynomials in
the variable x, of degree at most 1, with complex coeﬃcients. Suppose that ,·i is an
inner product on Vand β={x, x + 1 }is an orthonormal basis for V. Let T:VV
be the linear operator on Vdeﬁned by T(a+b x) = a+b+ (ai a+b)x,a,bC.
a) Find T(a+b x) for all a,bC.
b) For each cC, let Uc=T+cT . Find all complex numbers csuch that
9. Let T:VVbe a linear operator on a real inner product space V.
a) Let U=TT. Prove that U=Uand hU(x), xi= 0 for every vector xV.
b) Suppose that Vis ﬁnite-dimensional and has dimension at least 2. Using part
a), prove that there exists a nonzero linear operator U:VVsuch that
hU(x), xi= 0 for every vector xV.
10. Suppose that f(t) = antn+···+a1t+a0, where a0, . . . , anFand F=Ror F=C.
For T∈ L(V), deﬁne f(T) = anTn+···+a1T+a0IVand ¯
f(T) = ¯anTn+···+ ¯a1T+
¯a0IV. Assume that Vis a ﬁnite-dimensional inner product space. Let T∈ L(V) and
let U=f(T). Prove that U=¯
f(T).
11. §6.3, #13.
12. Let Tbe an invertible linear operator on a ﬁnite-dimensional inner product space V.
a) Prove that Tis invertible and (T)1= (T1).
b) Suppose that T=cT 1for some scalar cF. Show that Tis normal.
c) Show that if Tis as in part b), then λ¯
λ=cfor every eigenvalue λof 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 T=cT 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 T=cT 1if and only if for every
orthonormal basis {x1, . . . , xn}for V, the set {c1T(x1), . . . , c1T(xn)}is
also an orthonormal basis for V. (Here, cdenotes the positive square root of
c.)
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