6a. Suppose that B is a basis of V. Decompose it as a disjoint union B = B₁ I IIB₂ of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and V=W₁ 0... Ws is a direct sum of nonzero vector subspaces W₁ of V. 6b. Conversely, suppose that V = W₁ W, is a direct sum of nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that B = B₁ III B₂ is a basis of V and a disjoint union of non-empty sets B₁.

6a. Suppose that B is a basis of V. Decompose it as a disjoint union B = B₁ I IIB₂ of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and V=W₁ 0... Ws is a direct sum of nonzero vector subspaces W₁ of V. 6b. Conversely, suppose that V = W₁ W, is a direct sum of nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that B = B₁ III B₂ is a basis of V and a disjoint union of non-empty sets B₁.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 38EQ

See similar textbooks