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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 73CR

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Question

Let W and W, be two subspaces of a finite dimensional vector space V over a field F.
1. Show that W1n W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in
general.
2. Show that
dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2),
where W1 + W2 just denotes the span of W1 u W2 in V. (Hint: Apply the rank nullity theorem to the natural
map Wi O W2 → V.)

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