Math 1600A
Homework 9
Fall 2024
[5] 1. For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification.
(a) For any natural number m, the product of two non-zero numbers in Zm is non-zero.
(b) There are exactly 2 solutions to 4x = 2 in Z6.
(c) If T : R2 → R2 is a rotation about the origin such that T4 = I2, then T = I2 or T = -I2.
(d) Ifv ∈ Rn is an eigenvector of an n × n matrix A with eigenvalue 0, then rank(A) < n.
(e) If x is a steady state vector for a Markov chain with transition matrix P then
(P2 + 2P)(x) = 3x.
[3] 2. Consider the ISBN-10 code 3-5d0-96203-4. Recall that the check vector for ISBN-10 codes is
c = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1] ∈ Z 11(10).
Find the missing digit d. Show your work.
3. A Math 1600A student always submits their Gradescope assignments on Friday, Saturday or Sunday. If a student submits an assignment on a particular day of the week, then they will submit the next assignment on the same day 10% of the time, the later of the remaining two days 60% of the time and the remaining day 30% of the time.
[1] (a) Draw the state diagram of the associated Markov chain.
[1] (b) Find the transition matrix of the associated Markov chain.
[3] (c) If the student submits the first assignment on Friday, what is the probability that the student will submit the third assignment on Sunday?
[4] (d) What are the long-term probabilities of the student submitting an assignment on each of Friday, Saturday and Sunday, respectively?
4. Let T : R3 → R3 be a linear transformation determined by
[1] (a) Find the standard matrix of T.
[4] (b) Show that the standard matrix of T is invertible and find its inverse.
[1] (c) Find
5. Consider the matrix
[3] (a) Find the eigenvalues of A.
[4] (b) Find a basis for each eigenspace of A.