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Problem Set 3
Math 423, Section 200, Spring 2015
Due: Friday, February 13.
Review Chapter 4 and Section 5A 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 Friday, February 13. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. Let m be a non-negative integer. Determine whether the sets
{0} ∪ {p ∈ P(F) | deg(p) = m},
{0} ∪ {p ∈ P(F) | deg(p) = is even},
are subspaces of P(F).
2. Recall that RR is the real vector space of functions from R to R. Define T : P(R) → RR via


p(x) − p(3)



x,3
(T p)(x) = 
x−3


0
 p (3)
x=3
Show that, for all p ∈ P(R), T p is also a polynomial with real coefficients. Prove that T is a
linear map.
3. Let p ∈ P(C). Define q : C → C by q(z) = p(z)p( z ) (where the bars represent complex
conjugation). Show that q is a polynomial with real coefficients.
4. Let m be a non negative integer and p ∈ Pm (C). Suppose there exist distinct real numbers
x0 , x1 , . . . , xm such that p(x j ) ∈ R for j = 0, 1, . . . , m. Prove that p has real coefficients.
5. Find all the eigenvalues, eigenvectors and invariant subspaces of T : Fn → Fn given by
T (x1 , . . . , xn ) = (x1 , 2x2 , 3x3 , . . . , nxn ).
6. Find all eigenvalues and eigenvectors of T : P(R) → P(R) given by T p = p0 .
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7. Let T ∈ L (V), and suppose there exist nonzero v, w ∈ V such that T v = 3w and T w = 3v.
Show that either 3 or −3 is an eigenvalue of T .
8. Let U and W be nonzero subspaces of a vector space V, such that V = U ⊕ W. Define
P : V → V by T (u + w) = u for all u ∈ U and w ∈ W. Find all eigenvalues and eigenvectors
of P.
9. Let S , T ∈ L (V), and suppose S is invertible. Prove that T and S −1 T S have the same eigenvalues, and find the relationship between their respective eigenvectors.
10. Prove that T ∈ L (C∞ ) given by T (z1 , z2 , z3 , . . . ) = (0, z1 , z2 , z3 , . . . ) has no eigenvalues.
11. Compute all eigenvalues and eigenvectors of the backward shift operator T ∈ L (C∞ ) given
by T (z1 , z2 , z3 , . . . ) = (z2 , z3 , . . . ).
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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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