Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider the map Γ : W₁ × W₂ → W₁ + W₂ defined by I'(w₁, W₂) =W₁+ W₂. Prove the map is linear, and using I show that dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)
Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider the map Γ : W₁ × W₂ → W₁ + W₂ defined by I'(w₁, W₂) =W₁+ W₂. Prove the map is linear, and using I show that dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)
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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![Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider
the map I: W₁ × W₂ → W₁ + W₂ defined by I'(W₁, W₂) = W₁+w₂. Prove the map is
linear, and using I show that
dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fbd02ce7d-0410-41cf-a652-1995b327857b%2F5o8r47k_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider
the map I: W₁ × W₂ → W₁ + W₂ defined by I'(W₁, W₂) = W₁+w₂. Prove the map is
linear, and using I show that
dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)
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