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**Unformatted text preview: **MAT 442 December 2007 Linear Algebra Final Exam Qualifying Exam Write your answers on separate paper. Make sure your name is on every sheet. You may use any results from the course or text, except when asked to prove those results (which are marked as “result from course”). 1. (10 points each) Let V be the vector space of n × n matrices over a field F . (a) Prove that the trace, Tr( A ), gives a linear functional on V . (b) Let W be the set of matrices of trace 0. Prove that W is a subspace of V . (c) Determine the dimension of W , and prove your answer. 2. (10 points each) Suppose A is an n × n skew-symmetric matrix (i.e., A t =- A ) with entries from R , and that A is not the zero matrix. (a) Prove that if n is odd, then 0 is an eigenvalue for A . (b) Prove that A is diagonalizable over C . (c) Prove that A is not diagonalizable over R . 3. (10 points each) Let V be a finite dimensional vector space over a field F , and W 1 and W 2 subspaces of V ....

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