Practice Problems

Practice Problems - with reasoning based on theorems,...

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Math 136 Practice Problems # 2 1. Which of the following sets are linearly independent? If a set is linearly dependent, give a non-trivial linear combination of the vectors which equals the zero vector. a) S = 2 1 3 , 1 0 - 1 . b) T = 1 0 1 , - 2 0 - 2 , 1 1 1 . c) U = 0 0 0 , 1 1 - 1 , - 1 - 1 1 . 2. Determine, with proof, which of the following are subspaces of R 3 and which are not. a) S 1 = x 1 x 2 x 3 R 3 | x 1 = - x 2 b) S 2 = x 1 x 2 x 3 R 3 | x 1 - 2 x 2 + 3 x 3 = - 1 c) S 3 = x 1 x 2 x 3 R 3 | x 1 ,x 2 ,x 3 Z d) S 4 = x 1 x 2 x 3 R 3 | x 1 + x 2 = 0 ,x 3 = - x 2 e) S 5 = x 1 x 2 x 3 R 3 | x 1 + x 1 x 3 + x 2 = 0 1
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2 3. Suppose that the set of vectors { ~ v 1 , ~ v 2 , ~ v 3 ,..., ~ v k } is linearly independent, and that the set of vectors { ~ u 1 , ~ u 2 , ~ u 3 ,..., ~ u } is linearly dependent, prove that the set of vectors [ { 1 ~ v 1 , 2 ~ v 2 ,...,k~ v k , - 1 ~ u 1 , - 2 ~ u 2 ,..., - ‘~ u } is linearly dependent. 4. Determine whether each of the following statements is true or false. Justify your choices
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Unformatted text preview: with reasoning based on theorems, denitions, or counter-examples. a) If ~v 1 is a vector in R n , then the set { ~v 1 } is linearly independent. b) If { ~v 1 ,...,~v k } is a linearly dependent set in R n , then { ~v 1 ,...,~v k-1 } is linearly dependent. c) If { ~v 1 ,~v 2 } is a linearly independent set in R n , then { ~v 1 ,~v 1-~v 2 } is linearly independent. * 5. Let ~x = a b , ~ y = c d , and ~ z = e f be any three vectors in R 2 . a) Prove that { ~x,~ y } is linearly independent if and only if ad-bc 6 = 0. b) Prove that { ~x,~ y,~ z } is linearly dependent. * Indicates a challenging problem....
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This note was uploaded on 03/03/2011 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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Practice Problems - with reasoning based on theorems,...

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