ha3
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Linear algebra II Homework #3 due Thursday, Feb. 13 1. Suppose A, J are similar matrices and write J = B -1 AB for some matrix B. Given a basis v 1 , v 2 ,..., v k for the null space of J , find a basis for the null space of A (and also prove that it is a basis). 2. Find the Jordan form of each of the following matrices: A 1 = 0 2 3 3 1 1 -1 2 4 , A 2 = 0 2 3 -1 1 1 -1 2 4 . 3. Find the Jordan form of each of the following matrices: A 1 = 1 2 3 4 0 5 6 7 0 0 0 8 0 0 0 9 , A 2 = 2 0 2 1 0 2 1 2 0 0 1 1 0 0 0 1 . 4. Let A denote the n × n matrix whose entries are all equal to 1. Show that: (a) λ = 0 is an eigenvalue of A such that dim N (A - λI )= n - 1, (b) the first column of A is an eigenvector of A and A is diagonalizable. • When writing up solutions, write legibly and coherently. • Write your name and then MATHS/TP/TSM on the first page of your homework. • NO LATE HOMEWORK WILL BE ACCEPTED.
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Linear algebra IIHomework #3
due Thursday, Feb. 13
1. Suppose A; J are similar matrices and write J = B1AB for some matrix B. Givena basis v1;v2; : : : ;vk for the null space of J , nd a basis for the null space of A (andalso prove that it is a basis).
2. Find the Jordan form of each of the following matrices:
A1 =
24 0 2 33 1 11 2 4
35 ; A2 =24 0 2 31 1 11 2 4
35 :3. Find the Jordan form of each of the following matrices:
A1 =
26641 2 3 40 5 6 70 0 0 80 0 0 9
3775 ; A2 =26642 0 2 10 2 1 20 0 1 10 0 0 1
3775 :4. Let A denote the n n matrix whose entries are all equal to 1. Show that:
(a) = 0 is an eigenvalue of A such that dimN (A I) = n 1,(b) the rst column of A is an eigenvector of A and A is diagonalizable.
When writing up solutions, write legibly and coherently. Write your name and then MATHS/TP/TSM on the rst page of your homework. NO LATE HOMEWORK WILL BE ACCEPTED.
