# 4.8 Difference equations

2. (Verifying solutions.)

8. (Is a set of 3 vectors a basis for the solution space?) Verify that each of the 3 vectors is a solution. You need to show that the 3 vectors are LI.

18. (Find the general solution of a non-homogeneous difference equation.)

HINT: See hint for Exercise 17!

# 4.9 Markov chains

X. Find a stochastic matrix P with non-negative entries at least one of which is zero, for which all the entries of Pk are strictly positive for some k. In other words, find a regular stochastic matrix with a zero in it. Explicitly show that it is regular.

# 5.1 Eigenvectors and eigenvalues

32. (An eigenvalue/vector of a rotation about a line through origin.)

# 5.2 Characteristic Equation

16. (Eigenvalues and multiplicities for a diagonal matrix.)

22. (True/False and why.)

23. (RQ for A=QR.)

# 5.3 Diagonalization

2. (A4 from P,D.)

6. (Eigenvalues and bases for eigenspaces from diagonalization.)

22. (True/False and why.)