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Unformatted text preview: MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011, Key to HW#2 1. Determine if the following subsets of M 2 × 2 ( R ) is ars subspaces (you may assume that the operations on M 2 × 2 ( R ) are the usual addition and scalar multiplication. (a) ( a 1 a 2 a 4 ! ; a 1 ,a 2 ,a 4 ∈ R ) . (b) ( a 1 a 2 a 2 a 4 ! ; a 1 ,a 2 ,a 4 ∈ R ) . Solution (a) YES. it is closed under addition and scalar multiplication (you need to verity it, but I omit it here). (b) YES. it is closed under addition and scalar multiplication. (you need to verity it, but I omit it here). 2. Let W 1 ,W 2 be two subspaces of a vector space V . Prove that the inter section W 1 ∩ W 2 is also a subspace of V . Proof : Step 1: Let x,y be arbitrary two vectors in W 1 ∩ W 2 . Then x ∈ W 1 , x ∈ W 2 and y ∈ W 1 ,y ∈ W 2 . Since W 1 is a subspace, x + y ∈ W 1 . For the same reason, x + y ∈ W 2 . Hence x + y ∈ W 1 ∩ W 2 . So it is closed under addition....
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 Summer '08
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 Linear Algebra, Algebra, Addition, Multiplication, Scalar, Vector Space, Sets, W1 ⊂ W2

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