Download Math 314H Homework # 2 Due: Monday, April 1 Instructions: Do six

Math 314H
Homework # 2
Due: Monday, April 1
Instructions: Do six of the following seven problems. As before, if you do more than six
you will be graded on your best six. Some of the problems involve large matrices or require
you to compute large powers of a matrix, so you will no doubt want to use Maple or some
other technological aid. (Below I have listed a few Maple commands which might be useful for
this assignment.) Your solutions should clearly state where you have used any technological
assistance. You may want to include printouts of your Maple computations in the work you
turn in. (Be sure to label which computations refer to which problems.) Problems 5, 6, and 7
concern stochastic matrices and Markov chains. You should read section 4.9 in your text before
attempting these problems.
As before, your solutions should be written up neatly with correct grammar and punctuation.
You are free to use whatever resources you want and work with whomever you wish. However,
you must write up your solutions in your own words. Enjoy!
Here are some Maple commands which may be of use to you:
Command
evalm(Aˆ(-1));
evalm(Aˆ6);
nullspace(A);
What it does
returns the inverse of the matrix A
returns the 6th power of the matrix A
returns a basis for the null space of A
1. Let T : U → V be a linear transformation. Suppose {u1 , . . . , uk } is a basis for ker T
and {v1 , . . . , v` } a basis for im T . Now, as v1 , . . . , v` are in im T , there exists vectors
w1 , . . . , w` in U such that T (wi ) = vi for i = 1, . . . , n. Prove that {u1 . . . , uk , w1 , . . . w` }
spans U . (This completes the proof of the theorem from class on March 13.)
2. Prove that the set of functions β = {1, cos x, cos2 x, . . . , cos6 x} is linearly independent in
C(R). (Hint: Suppose c0 · 1 + c1 · cos x + · · · + c6 · cos6 x = 0. Then this equation holds
for all values of x. Hence, for each value you substitute in for x, you get a different linear
equation in the ci ’s. Use different values for x until you get enough independent equations
to force all the ci ’s to be zero.)
3. Let W = Span(β) where β is the set of functions from the previous problem. Then β is a
basis for W (as β is linearly independent and spans W ). Let α = {1, cos x, cos 2x, . . . , cos 6x}.
Prove that α is also a basis for W . You may assume the following trig identities:
cos 2x = −1 + 2 cos2 x
cos 3x = −3 cos x + 4 cos3 x
cos 4x = 1 − 8 cos2 x + 8 cos4 x
cos 5x = 5 cos x − 20 cos3 x + 16 cos5 x
cos 6x = −1 + 18 cos2 x − 48 cos4 x + 32 cos6 x.
4. Let W, α, β be as in the previous problem.
(a) Find the change of coordinates matrix from α to β.
(b) Find the change of coordinates matrix from β to α. (Hint: See the gray box on page
268 of your text.)
(c) Use the matrix you found in (b) to rewrite the following indefinite integral as an
integral of a linear combination of functions from α:
Z
7 + 3 cos2 x − 7 cos4 x + 5 cos5 x − 13 cos6 x dx.
(d) Evaluate (by hand) the integral you found in part (c).
5. Let P be an n × n matrix all of whose entries are nonnegative. Let S be the n × n matrix
which has a ‘1’ in every entry.
(a) Prove that P is a stochastic matrix if and only if SP = S.
(b) Use part (a) to prove that if P and Q are stochastic matrices, so is P Q.
(c) Prove that if P is a stochastic matrix, so is P n for every n ≥ 1.
(d) If P is an invertible stochastic matrix, is P −1 necessarily stochastic? Justify your
answer.
6. In this exercise you will prove that every stochastic matrix has at least one steady state
vector. Let P be an n × n stochastic matrix.
(a) Let I be the n × n identity matrix. Prove that P − I is not invertible. (Hint: From
the preceding problem, you know that S(P − I) = 0.)
(b) Prove there exists a nonzero vector u such that P u = u. (Hint: Use the theorem on
page 262.)
(c) Show that there exists a probability vector q such that P q = q.
7. A mouse is placed in a box with nine rooms as shown in the figure below. Every 10
seconds, an observer checks to see which room the mouse is in. Assume that for each 10
second interval it is equally likely that the mouse goes through any door in the room, or
stays put in the room.
(a) Suppose the mouse is initially placed in room 1. What is the probability the mouse
will be in room 9 one minute later?
(b) Find the steady state vector for this Markov chain.