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Let W1 and W2 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 W¡u W2 need not be a vector subspace in general.

Let W1 and W2 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 W¡u W2 need not be a vector subspace in general.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.)
Transcribed Image Text: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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