hw3s(1) - MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011,...

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MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011, Key to HW#3 1. Problem 2 (e), (f), (i) on Page 41. Solution : (e) Write a 1 ((1 , - 1 , 2) + a 2 (1 , - 2 , 1) + a 3 (1 , 1 , 4) = (0 , 0 , 0) . By solving the system of equations a 1 + a 2 + a 3 = 0 , - a 1 - 2 a 2 = 0 , 2 a 1 + a 2 + 4 a 3 = 0 we get a 1 = - 3 ,a 2 = 2 ,a 3 = 1. So they are linearly dependent. (f) Write a 1 ((1 , - 1 , 2) + a 2 (2 , 0 , 1) + a 3 ( - 1 , 2 , - 1) = (0 , 0 , 0) . By solving the system of equations, we get a 1 = a 2 = a 3 = 0. So it is linearly independent. (i) Write a 1 ( x 4 - x 3 +5 x 2 - 8 x +6)+ a 2 ( - x 4 + x 3 - 5 x 2 +5 x - 3)+ a 3 ( x 4 +3 x 2 - 3 x +5) + a 4 (2 x 4 + 3 x 3 + 4 x 2 - x + 1) + a 5 ( x 3 - x + 2) = 0 . So we get system of equations a 1 - a 2 + a 3 + 2 a 4 = 0 ,.... , 6 a 1 - 3 a 2 + 5 a 3 + a 4 + 2 a 5 = 0 . By solving this, we get a 1 = ··· = a 5 = 0. Hence it is linearly independent. 2. Problem 3 on Page 41. Solution : Omitted. 3. Problem 13 on Page 42. Solution : (a) This is a ”two-way” statement. We first prove ”= ”, i.e. assume that { u,v } is linearly independent, we prove that
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.

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hw3s(1) - MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011,...

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