Page 170 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
158
Exercises
1. Make P 2 (R) into an inner-product space by defining
1
p, q = p(x)q(x) dx.
0
2
Define T ∈L(P 2 (R)) by T(a 0 + a 1 x + a 2 x ) = a 1 x.
(a) Show that T is not self-adjoint.
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(b) The matrix of T with respect to the basis (1,x,x ) is
 
0 0 0
 0 1 0  .
 
0 0 0
This matrix equals its conjugate transpose, even though T
is not self-adjoint. Explain why this is not a contradiction.

2. Prove or give a counterexample: the product of any two self-
adjoint operators on a finite-dimensional inner-product space is
self-adjoint.

3. (a) Show that if V is a real inner-product space, then the set
of self-adjoint operators on V is a subspace of L(V).
(b) Show that if V is a complex inner-product space, then the
set of self-adjoint operators on V is not a subspace of
L(V).
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4. Suppose P ∈L(V) is such that P = P. Prove that P is an orthog-
onal projection if and only if P is self-adjoint.
5. Show that if dim V ≥ 2, then the set of normal operators on V is
not a subspace of L(V).

6. Prove that if T ∈L(V) is normal, then
range T = range T .
∗

7. Prove that if T ∈L(V) is normal, then
k
k
null T = null T and range T = range T
for every positive integer k.