Exercise 1.9 Consider the following subspaces of R³: V₁ V₂ = ((1,0, 1, 1, 0), (-1, 1, 0, 0, 1), (0, 0, 0, 1, -1)) ((0, 1, 1, 1, 1), (1, 1, 2, 2, 1), (1, 0, 1, 1, 1)) a) Determine a basis B₁ of V₁ and a basis B₂ of V₂. Compute dim(V₁) and dim(V₂). b) Determine a basis of V₁ V₂ and one of V₁ + V₂. Compute dim(V₁V₂) and dim(V₁+V₂). Are the subspaces V₁ and V₂ in direct sum? c) Write if possible the vector v = (1,0, 1, 1, 0) as a sum v = V₁ + V₂ with v₁ € V₁, V2 € V₂ and v₁ (0,0,0,0,0) and v2 # (0, 0, 0, 0, 0). Is the expression unique?
Exercise 1.9 Consider the following subspaces of R³: V₁ V₂ = ((1,0, 1, 1, 0), (-1, 1, 0, 0, 1), (0, 0, 0, 1, -1)) ((0, 1, 1, 1, 1), (1, 1, 2, 2, 1), (1, 0, 1, 1, 1)) a) Determine a basis B₁ of V₁ and a basis B₂ of V₂. Compute dim(V₁) and dim(V₂). b) Determine a basis of V₁ V₂ and one of V₁ + V₂. Compute dim(V₁V₂) and dim(V₁+V₂). Are the subspaces V₁ and V₂ in direct sum? c) Write if possible the vector v = (1,0, 1, 1, 0) as a sum v = V₁ + V₂ with v₁ € V₁, V2 € V₂ and v₁ (0,0,0,0,0) and v2 # (0, 0, 0, 0, 0). Is the expression unique?
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...
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Transcribed Image Text:Exercise 1.9 Consider the following subspaces of R5:
V₁=
V₂ =
((1,0, 1, 1, 0), (-1, 1, 0, 0, 1), (0, 0, 0, 1, -1))
((0, 1, 1, 1, 1), (1, 1, 2, 2, 1), (1, 0, 1, 1, 1))
a) Determine a basis B₁ of V₁ and a basis B2 of V₂. Compute dim(V₁) and dim(V₂).
b) Determine a basis of V₁ V₂ and one of V₁ + V₂. Compute dim(V₁V₂) and dim(V₁+V₂).
Are the subspaces V₁ and V₂ in direct sum?
c) Write if possible the vector v = (1, 0, 1, 1, 0) as a sum v = V₁ + V₂ with v₁ € V₁, V₂ € V₂
and v₁ (0,0,0,0,0) and v₂ # (0, 0, 0, 0, 0). Is the expression unique?
d) Determine, if it exists, a subspace T ≤ R5 such that (TV₁) V₂ = (TV₂) V₁ =
V₁ + V₂.
e) Determine S < R5 such that S (V₁ V₂) = V₁ + V₂. Is it unique?
f) For each possible choice of S≤ R5 such that S (V₁ V₂) = V₁+V₂, compute dim(SV₁)
and dim(SnV₂).
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