Final Exam - Answer Key to Final Examination Version 1 Math...

Answer Key to Final Examination, Version 1, Math. 33A, Spring Quarter, 2003
1. Suppose matrix A is similar to B . Prove the following facts (missing step by step
description of your reasoning costs maximum a half of the allocated points):
(a) There exists an isomorphism from the kernel of A onto the kernel of B (50 points).
Since A is similar to B , we have an invertible matrix S such that B SAS
1
BS
SA S
1
B AS
1
. Then if x is in the kernel of A , then BS x SA x S 0 0 . This
shows S x is in the kernel of B . Thus we have a linear transformation T : Ker A Ker B
given by T x S x . Since S is invertible, Ker T Ker S 0 ; so, Ker T 0 .
If y is in the kernel of B , AS
1
y S
1
B y S
1
0 0 . Thus S
1
y is in the
kernel of A . In particular y SS
1
y is in the image of T . This shows that Im T
Ker B . The two facts Ker T 0 and Im T Ker B combined tell us that T is an
isomorphism, and T
1
is given by the multiplication by S
1
.
(b) nullity A nullity B (20 points).
By (a), Ker A and Ker B has the same dimension, which are the nullity of A and B ,
respectively.