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Problem Set 9
Math 423, Section 200, Spring 2015
Due: Monday, April 20.
Review Sections 7C, 7D and 8A in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Monday, April 20. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. Prove or give a counterexample: if T ∈ L (V) is self-adjoint and there exists an orthonormal
basis e1 , . . . , en of V such that hT ei , ei i ≥ 0 for i = 1, . . . , n then T is positive.
2. Suppose T is a positive operator on a finite dimensional inner product space V. Suppose
v, w ∈ V are such that T v = w and T w = v. Show that v = w.
3. Let V be a finite dimensional inner product space. Prove that if T ∈ L (V) is positive, then T k
is positive for every positive integer k.
4. Let V be a finite dimensional inner product space. Prove that if S , T ∈ L (V) are isometries,
then S T is an isometry.
5. Fix z ∈ C and define T z : C → C by T z (w) = zw. Characterize for which z the operator T z is:
• normal,
• self-adjoint,
• an isometry.
6. Note that Definition 7.37 in the book makes sense for infinite dimensional inner product
spaces. Let V be the C-vector space of continuous functions from [0, 1] to C with inner product
Z 1
h f, gi =
f (t)g(t)dt.
0
Let h ∈ V and define T ∈ L (V) by T f = h f . Show that T is an isometry if and only if
|h(t)| = 1 for all t ∈ [0, 1].
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√
√
7. Let T : C2 → C2 defined by T (z1 , z2 ) = ((1 − −1)z2 , (1 + −1)z1 + z2 ). Find the singular
values s1 ≥ s2 of T , and find orthonormal bases v1 , v2 and u1 , u2 of C2 such that T vi = si ui .
8. Find
for T : R3 → R3 whose matrix with respect to the standard basis
 a polar decomposition

20 4 0


is:  0 0 1.


4 20 0
9. Let V be a finite dimensional inner product space. Prove that T ∈ L (V) is invertible if and
only if 0 is not a singular value of T .
10. Let V be a finite dimensional inner product space, and let T ∈ L (V). Prove that the dimension
of range T equals the number of nonzero singular values of T .
11. Let V be a finite dimensional inner product space. Prove that T ∈ L (V) is an isometry if and
only if all the singular values of T equal 1.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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