Homework 9
Problem 1. Let A and B be two similar matrices. Prove that Ak is also similar to B k .
Problem 2. The matrix A is factored in the form P DP 1 . Use
the eigenvalues and corresponding eigenvectors of A.
3 0 0
A = 3 4 9
0 0 3
3 0 1
3 0 0
0
0 1 3
Homework 10
2
1
1
Problem 1. Let H be a subspace spanned by u1 = 1 and u1 = 3 . Write y = 2
3
2
1
as the sum of a vector in H and a vector orthogonal to H.
Problem 2. Find the closest point to y in the subspace spanned by v1 and v2 , where
4
1
1
2
0
Homework 7
Problems for Section 2.8 and 2.9
Problem 1. Suppose A is m n. Prove the following equality
dim Col(A) + dim N ul(AT ) = m
Problem 2. Suppose A is m n and b is in Rm . Prove that if the equation Ax = b is consistent,
then rank [A, b] = rank A.
P
Homework 8
Problem 1. Determine whether v is an eigenvector of the given matrix.
3 5 1
1
9
1 , v = 1
A = 1
1
5
5
4
1
1
1
3
4 .
Problem 2. Determine if = 1 is an eigenvalue of 2
2 3 4
Problem 3. Find the eigenvalues and associated eigenvectors for the fol
Homework 6
Problems for Section 2.8 and 2.9
1
2
3
Problem 1. Let v1 = 3 , v2 = 3 , and w = 3 . Determine if w is in the subspace
4
7
10
spanned by v1 and v2 .
2
2
0
6
Problem 2. Let v1 = 0 , v2 = 3 , v3 = 5 , and w = 1 . Determine if w is
6
3
5
17
in Col