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 W₁ U W₂ need not be a vector subspace in general. 2. Show that dim(W₁) + dim(W₂) – dim(W₁ ʼn 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 W₁ W₂ → V.)
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 W₁ U W₂ need not be a vector subspace in general. 2. Show that dim(W₁) + dim(W₂) – dim(W₁ ʼn 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 W₁ W₂ → V.)
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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Transcribed Image Text: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 W₁ U W₂ need not be a vector subspace in
general.
2. Show that
dim(W₁) + dim(W₂) – dim(W₁n 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 W₁ W₂ → V.)
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