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(d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³ with the standard operations. Justify your answer.

(d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³ with the standard operations. Justify your answer.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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answer part d

(a) Given u =(3,-1,2,3) and v= (0,2,2,1) in R¹. Find 3u-2v. =(4,4,5) as a linear combination of u, = (1,2,3), u, = (-2,0,1) and u, = (1,0,0). (b) Write v=(4,4,5) (c) Determine whether the set R², with the operations (x₁, y₁) + (x₂, y₂ ) = (x,,0) and c(x, y) = (x,cy) is a vector space. If it is, verify each vector space axiom. If not, state all vector space axioms that fail. (d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³ with the standard operations. Justify your answer. (e) Determine whether the set S = {x²-2x,x² +8,x¹-x²₁x² -4} spans P₁.
Transcribed Image Text:(a) Given u =(3,-1,2,3) and v= (0,2,2,1) in R¹. Find 3u-2v. =(4,4,5) as a linear combination of u, = (1,2,3), u, = (-2,0,1) and u, = (1,0,0). (b) Write v=(4,4,5) (c) Determine whether the set R², with the operations (x₁, y₁) + (x₂, y₂ ) = (x,,0) and c(x, y) = (x,cy) is a vector space. If it is, verify each vector space axiom. If not, state all vector space axioms that fail. (d) Determine whether the set W = {(x₁,x₂,4): x, and x₂ are real numbers is a subspace of R³ with the standard operations. Justify your answer. (e) Determine whether the set S = {x²-2x,x² +8,x¹-x²₁x² -4} spans P₁.
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