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Prove that if W₁ and W₂ are finite-dimensional subspaces of a vector space V, then W₁ + W₂ is finite-dimensional, and dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ n W₂) - ... Hint: Let B₁ = (u₁,, uk) be a basis of W₁ n W2, hence dim W₁n W₂ k. Extending Bo into B₁ = (u1, ***, Uk, V1, ,U₁), a basis of W₁, we have dim W₁ k+ 1. Extending Bo into B2 = (u1, ..., uk, P1, * Pm), a basis of W2, we have dim W₂ = k + m. Therefore, it suffices to show that = = dim(W₁ + W₂) = k +l+m since dim(W₁) + dim(W₂) - dim(W₁ n W₂) = (k + 1) + (k + m) − k = k +l+m. Now let B = (u₁,· ···‚ Uk, U1, ***, VI, P1, …, pm). Show that B is a basis of W₁ + W₂ by proving: " 1. B spans W₁ + W₂ 2. B is linearly indepedent. 2

Prove that if W₁ and W₂ are finite-dimensional subspaces of a vector space V, then W₁ + W₂ is finite-dimensional, and dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ n W₂) - ... Hint: Let B₁ = (u₁,, uk) be a basis of W₁ n W2, hence dim W₁n W₂ k. Extending Bo into B₁ = (u1, ***, Uk, V1, ,U₁), a basis of W₁, we have dim W₁ k+ 1. Extending Bo into B2 = (u1, ..., uk, P1, * Pm), a basis of W2, we have dim W₂ = k + m. Therefore, it suffices to show that = = dim(W₁ + W₂) = k +l+m since dim(W₁) + dim(W₂) - dim(W₁ n W₂) = (k + 1) + (k + m) − k = k +l+m. Now let B = (u₁,· ···‚ Uk, U1, ***, VI, P1, …, pm). Show that B is a basis of W₁ + W₂ by proving: " 1. B spans W₁ + W₂ 2. B is linearly indepedent. 2

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
Section4.5: Basis And Dimension
Problem 69E: Find a basis for R2 that includes the vector (2,2).
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Prove that if W₁ and W₂ are finite-dimensional subspaces of a vector space V, then W₁ + W₂ is finite-dimensional, and dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ n W₂) - ... Hint: Let B₁ = (u₁,, uk) be a basis of W₁ n W2, hence dim W₁n W₂ k. Extending Bo into B₁ = (u1, ***, Uk, V1, ,U₁), a basis of W₁, we have dim W₁ k+ 1. Extending Bo into B2 = (u1, ..., uk, P1, * Pm), a basis of W2, we have dim W₂ = k + m. Therefore, it suffices to show that = = dim(W₁ + W₂) = k +l+m since dim(W₁) + dim(W₂) - dim(W₁ n W₂) = (k + 1) + (k + m) − k = k +l+m. Now let B = (u₁,· ···‚ Uk, U1, ***, VI, P1, …, pm). Show that B is a basis of W₁ + W₂ by proving: " 1. B spans W₁ + W₂ 2. B is linearly indepedent. 2
Transcribed Image Text:Prove that if W₁ and W₂ are finite-dimensional subspaces of a vector space V, then W₁ + W₂ is finite-dimensional, and dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ n W₂) - ... Hint: Let B₁ = (u₁,, uk) be a basis of W₁ n W2, hence dim W₁n W₂ k. Extending Bo into B₁ = (u1, ***, Uk, V1, ,U₁), a basis of W₁, we have dim W₁ k+ 1. Extending Bo into B2 = (u1, ..., uk, P1, * Pm), a basis of W2, we have dim W₂ = k + m. Therefore, it suffices to show that = = dim(W₁ + W₂) = k +l+m since dim(W₁) + dim(W₂) - dim(W₁ n W₂) = (k + 1) + (k + m) − k = k +l+m. Now let B = (u₁,· ···‚ Uk, U1, ***, VI, P1, …, pm). Show that B is a basis of W₁ + W₂ by proving: " 1. B spans W₁ + W₂ 2. B is linearly indepedent. 2
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