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assign2_soln

# assign2_soln - Math 136 Assignment 2 Solutions 1 Determine...

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Math 136 Assignment 2 Solutions 1. Determine, with proof, which of the following are subspaces of R 3 and which are not. a) S 1 = x 1 0 x 2 R 3 | x 1 - x 2 = 0 Solution: By definition S 1 is a subset of R 3 . Also, 0 0 0 S 1 since 0 - 0 = 0. Thus, S 1 is a non-empty subset of R 3 , so we can apply the Subspace Test. Let ~x = x 1 0 x 2 , ~ y = y 1 0 y 2 S 1 . Then x 1 - x 2 = 0 and y 1 - y 2 = 0. Then we have ~x + ~ y = x 1 + y 1 0 x 2 + y 2 and ( x 1 + y 1 ) - ( x 2 + y 2 ) = x 1 - x 2 + y 1 - y 2 = 0 - 0 = 0 So, ~x + ~ y satisfies the condition of S 1 , so ~x + ~ y S 1 . Similarly, c~x = cx 1 0 cx 2 and cx 1 - cx 2 = c ( x 1 - x 2 ) = c (0) = 0 so c~x S 1 . Thus, by the Subspace Test, S 1 is a subspace of R 3 . b) S 2 = x 1 x 2 x 3 R 3 | x 1 , x 2 , x 3 Z Solution: The definition of R n allows multiplication by any real scalar. Thus, the fact that we are restricting the entries of the vectors in S 2 to be integers makes us believe that the set is not a subspace. For example if c = 2 and ~x = 1 1 1 , then c~x = 2 2 2 6∈ S 2 . Therefore, S 2

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