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assign2_soln

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

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Assignment 2 Solutions 1. Determine, with proof, which of the following are subspaces of R 3 and which are not. a) S 1 = { ~x R 3 | x 1 - x 2 = x 3 } 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 an apply the Subspace Test. Let ~x = x 1 x 2 x 3 , ~ y = y 1 y 2 y 3 S 1 . Then x 1 - x 2 = x 3 and y 1 - y 2 = y 3 . This gives ~x + ~ y = x 1 + y 1 x 2 + y 2 x 3 + y 3 and ( x 1 + y 1 ) - ( x 2 + y 2 ) = x 1 - x 2 + y 1 - y 2 = x 3 - y 3 So, ~x + ~ y satisﬁes the condition of S 1 , so ~x + ~ y S 1 . Similarly, c~x = cx 1 cx 2 cx 3 and cx 1 - cx 2 = c ( x 1 - x 2 ) = cx 3 so c~x S 1 . Thus, by the Subspace Test, S 1 is a subspace of R 3 . b) S 2 = 1 a b | a,b R Solution: By deﬁnition, S 2 is a subset of R 3 , but 0 0 0 6∈ S 2 . Therefore, S 2 is not a subspace. 1

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

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