# hw0 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 0:...

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Unformatted text preview: Math508, Fall 2010 Jerry L. Kazdan Problem Set 0: Rust Remover DUE: These problems will not be collected. You should already have the techniques to do these problems, although they may take some thinking. 1. Show that for any positive integer n , the number 2 n + 2 + 3 2 n + 1 is divisible by 7. 2. Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y . Give a proof or counterexample for each of the following. a) If n = k there is always at most one solution. b) If n &amp;gt; k you can always solve AX = Y . c) If n &amp;gt; k the nullspace of A has dimension greater than zero. d) If n &amp;lt; k then for some Y there is no solution of AX = Y . e) If n &amp;lt; k the only solution of AX = 0 is X = 0. 3. Let A and B be n n matrices with AB = 0. Give a proof or counterexample for each of the following. a) BA = b) Either A = 0 or B = 0 (or both). c) If det A =- 3, then B = 0. d) If B is invertible then A = 0....
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## This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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hw0 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 0:...

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