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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 4 If V is a vector space over F, we will write dimF (V ) for the dimension of V when we wish to emphasize the field of scalars. For example, dimC (C3 ) = 3 and dimR (C3 ) = 6. 1. Let W1 , W2 be subspaces of V such that V = W1 ⊕ W2 . Let W be a subspace of V . Show that if W1 ⊆ W or W2 ⊆ W , then W = (W ∩ W1 ) ⊕ (W ∩ W2 ). Is this still true if we omit the condition ‘W1 ⊆ W or W2 ⊆ W ’ ? 2. For the following vector spaces V , find the coordinate representation of the respective elements. (a) V = P2 = {ax2 + bx + c | a, b, c ∈ R}. Find [p(x)]B where p(x) = 2x2 − 5x + 6, B = [1, x − 1, (x − 1)2 ]. (b) V = R2×2 . Find [A]B where 2 3 1 1 0 −1 1 −1 1 0 A= , B= , , , . 4 −7 1 1 1 0 0 0 0 0 (c) V = R2 . Let θ ∈ R be fixed. Find [v]B where x cos θ − sin θ v= , B= , . y sin θ cos θ 3. Let W1 and W2 be the following subspaces of R4 . 1 4 5 1 2 2 −1 1 −1 −1 , , , W2 = span , . W1 = span 3 3 6 1 4 6 6 12 1 5 (a) (b) (c) (d) (e) Find a basis of W1 ∩ W2 . Find a basis of W1 + W2 . Extend the basis of W1 ∩ W2 in (a) to get a basis of W1 . Extend the basis of W1 ∩ W2 in (a) to get a basis of W2 . From the bases in (c) and (d), obtain a basis of W1 + W2 . 4. Let W1 , W2 , W3 be subspaces of a vector space V . (a) Show that dim(W1 + W2 ) + dim(W1 ∩ W2 ) = dim(W1 ) + dim(W2 ). (b) Suppose dim(W1 + W2 ) = dim(W1 ∩ W2 ) + 1. Show that either W1 ⊆ W2 or W2 ⊆ W1 . (c) Show that dim(W1 ∩ W2 ∩ W3 ) ≥ dim(W1 ) + dim(W2 ) + dim(W3 ) − 2 dim(V ). Date: March 6, 2008 (Version 1.0); due: March 13, 2008. 1 5. Let V be a vector space over R. We have seen in Homework 1 Problem 2 that W = V × V may be made into a vector space over C with appropriate addition and scalar multiplication. W is called the complexification of V . Show that dimC (W ) = dimR (V ). 2