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hw3 - Math 260 Spring 2012 Jerry L Kazdan Problem Set 3 Due...

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Math 260, Spring 2012 Jerry L. Kazdan Problem Set 3 Due: In class Thursday, Feb. 2. Late papers will be accepted until 1:00 PM Friday . 1. 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 > k you can always solve AX = Y . c) If n > k the nullspace of A has dimension greater than zero. d) If n < k then for some Y there is no solution of AX = Y . e) If n < k the only solution of AX = 0 is X = 0. 2. Let A and B be n × n matrices with AB = 0. Give a proof or counterexample for each of the following. a) BA = 0 b) Either A = 0 or B = 0 (or both). c) If B is invertible then A = 0. d) There is a vector V = 0 such that BAV = 0. 3. Consider the system of equations x + y - z = a x - y + 2 z = b. a) Find the general solution of the homogeneous equation. b) A particular solution of the inhomogeneous equations when a = 1 and b = 2 is x = 1 , y = 1 , z = 1. Find the most general solution of the inhomogeneous equations. c) Find some particular solution of the inhomogeneous equations when a = - 1 and b = - 2. d) Find some particular solution of the inhomogeneous equations when a = 3 and b = 6. [Remark: After you have done part a), it is possible immediately to write the solutions to the remaining parts.] 4. Let A = 1 1 - 1 1 - 1 2 . a) Find the general solution Z of the homogeneous equation A Z = 0.
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