Math110s-hw3 - u 1 u m ⊕ span v 1 v n(c Suppose u 1 u m are linearly independent but u 1 u m v are linearly dependent Show that v ∈ span u 1 u

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 3 1. Let W 1 ,...,W n be non-trivial subspaces of a vector space V over R . Show that there exists a vector v V such that v / W i for all i = 1 ,...,n . 2. Let W 1 ,...,W n be subspaces of a vector space V . Recall from Problem Set 2 that W 1 + ··· + W n is a direct sum (and hence may be denoted W 1 ⊕ ··· ⊕ W n ) if W i ± j 6 = i W j ² = { 0 } for all i = 1 ,...,n. Show that the following statements are equivalent. (i) W 1 + ··· + W n is a direct sum. (ii) The function defined by f : W 1 × ··· × W n V, f ( w 1 ,..., w n ) = w 1 + ··· + w n is injective. (iii) W 1 ,...,W n satisfies ( W 1 + ··· + W i ) W i +1 = { 0 } for all i = 1 ,...,n - 1 . 3. Let V be a vector space and u 1 ,..., u n V . (a) Suppose v 1 ,..., v m span { u 1 ,..., u n } . Show that span { u 1 ,..., u n , v 1 ,..., v m } = span { u 1 ,..., u n } . (b) Suppose u 1 ,..., u m , v 1 ,..., v n are linearly independent. Show that span { u 1 ,..., u m , v 1 ,..., v n } = span
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Unformatted text preview: { u 1 ,..., u m } ⊕ span { v 1 ,..., v n } . (c) Suppose u 1 ,..., u m are linearly independent but u 1 ,..., u m , v are linearly dependent. Show that v ∈ span { u 1 ,..., u m } and v can be uniquely expressed as a linear combination of u 1 ,..., u m . 4. Let V be a vector space and u 1 ,..., u n ∈ V . For all i = 1 ,...,n , let W i = span { u 1 ,..., u i } and let W = { } . Show that u 1 ,..., u n are linearly independent if and only if W i-1 ( W i for i = 1 ,...,n. Recall that the notation A ( B means two things: A ⊆ B and A 6 = B . Date : February 28, 2008 (Version 1.1); due: March 6, 2008. 1...
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This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.

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