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

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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.
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