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Let W₁, W2, ..., We are subspaces of a finite-dimensional vector space V such that V=W₁ +W₂ + ... + Wk. Let 3; denote a basis for W, and let 3 = and only if ß is a basis for V. UB. Prove that V = W₁ W₂2 ... Wk if O i=1

Let W₁, W2, ..., We are subspaces of a finite-dimensional vector space V such that V=W₁ +W₂ + ... + Wk. Let 3; denote a basis for W, and let 3 = and only if ß is a basis for V. UB. Prove that V = W₁ W₂2 ... Wk if O i=1

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 33EQ
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12. Let W₁, W2, We are subspaces of a finite-dimensional vector space V such that ...2 V=W₁ + W₂ + ... + Wk. k Let denote a basis for W; and let 3 = UB₁. Prove that V = W₁ W₂ ... and only if ß is a basis for V. i=1 Wk if
Transcribed Image Text:12. Let W₁, W2, We are subspaces of a finite-dimensional vector space V such that ...2 V=W₁ + W₂ + ... + Wk. k Let denote a basis for W; and let 3 = UB₁. Prove that V = W₁ W₂ ... and only if ß is a basis for V. i=1 Wk if
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