554-HW-2q - Hoffman and Kunze - Linear Algebra MA 530 - HW...

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Unformatted text preview: Hoffman and Kunze - Linear Algebra MA 530 - HW 2: p40/ 5,7,8,9; p48/ 7,8,9,13; p54/ 1,3; p66/ 3, 7 p40, 7 We have subspaces W1 , W2 ⊆ V , for V a vectorspace, such that W1 ∪ W2 is a subspace of V . Show one of W1 , W2 is contained in the other. p40, 8 V is the vectorspace of all functions from R into R. Let Ve be the subset of V of all even functions and Vo the subset of all odd functions. Show (a) each is a subspace of V , (b) Ve + Vo = V , and (c) they have trivial intersection. p40, 9 Let W1 and W2 be subspaces of V such that V = W1 + W2 and W1 ∩ W2 = {0}. Prove for each vector v ∈ V that there are unique vectors a ∈ W1 and b ∈ W2 such that v = a + b. p48, 8 Find a basis {A1 , A2 , A3 , A4 } for F 2×2 such that each Ai satisfies A2 = Ai . i p48, 9 Given: x, y, z are linearly independent vectors in V a vectorspace. Show that (x + y ), (y + z ), (z + x) are also linearly independent in V , a vectorspace over C. p48, 13 Let V be a vectorspace over F2 . Given: x, y, z are linearly independent vectors in V . Discuss whether (x + y ), (y + z ), (z + x) are also linearly independent in V . p54, 3 Let b1 = (1, 0, −1), b2 = (1, 1, 1), b3 = (0, 0, 1) be an ordered basis for R3 . Find the coordinate matrix C of (x, y, z ) an arbitrary vector in R3 relative to the ordered basis given above. p66, 3 Consider the vectors v1 = (−1, 0, 1, 2); v2 = (3, 4, −2, 5); v3 = (1, 4, 0, 9). Find a system of homogeneous linear equations for which the space of solutions is exactly the subspace of R4 spanned by the three given vectors. p66, 7 Let A ∈ F m×n and consider the system of equations AX = Y . Prove that this system of equations has a solution iff the row rank of A = the row rank of [A : Y ] i.e. the augmented matrix of the system. 1 ...
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This document was uploaded on 01/25/2012.

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