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260S12Ex1s - Math 260 Feb 9 2012 Exam 1 Jerry L Kazdan...

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Math 260 Exam 1 Jerry L. Kazdan Feb. 9, 2012 12:00 – 1:20 Directions This exam has 10 questions (10 points each). Closed book, no calculators or computers– but you may use one 3 × 5 card with notes on both sides. Neatness counts . 1. Which of the following sets are linear spaces? a) The points X = ( x 1 , x 2 , x 3 ) in R 3 with the property x 1 - 2 x 3 = 0. b) The set of solutions x of Ax = 0, where A is an m × n matrix. c) The set of polynomials p ( x ) with 1 - 1 p ( x ) cos 2 x dx = 0. d) The set of solutions y = y ( t ) of y + 4 y + y = x 2 - 3. [ Note: You are not being asked to solve this differential equation. You are only being asked a more primitive question.] 2. Let S and T be linear spaces and L : S T be a linear map. Say V 1 and V 2 are (distinct!) solutions of the equations LX = Y 1 while W is a solution of LX = Y 2 . Answer the following in terms of V 1 , V 2 , and W . a) Find some solution of LX = 2 Y 1 - 3 Y 2 . b) Find another solution (other than W ) of LX = Y 2 . 3. Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y .
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