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In each of the following determine whether the set U is a subspace of the given vector space V. (a) V = R², U = {(x, y) = R² | x+y=1}, (b) V=R, U = Z, (c) V = Mat2,2 (R), U = {A E Mat2,2 (R) | trace (A) = 0}, V = Mat2,2 (R), U = {A E Mat2,2 (R) | det (A)=0}, V = P3(R), U = {f E P3(R) | f(x²) = f(x)²}. (d) (e) 1 2 (²) = ¹ 34 Note: The trace of a matrix is the sum of its diagonal entries e.g. trace = 1+4= 5.

In each of the following determine whether the set U is a subspace of the given vector space V. (a) V = R², U = {(x, y) = R² | x+y=1}, (b) V=R, U = Z, (c) V = Mat2,2 (R), U = {A E Mat2,2 (R) | trace (A) = 0}, V = Mat2,2 (R), U = {A E Mat2,2 (R) | det (A)=0}, V = P3(R), U = {f E P3(R) | f(x²) = f(x)²}. (d) (e) 1 2 (²) = ¹ 34 Note: The trace of a matrix is the sum of its diagonal entries e.g. trace = 1+4= 5.

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
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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1. In each of the following determine whether the set U is a subspace of the given vector space V. V = R², U = {(x, y) = R² | x+y=1}, V=R, U = Z, (b) (d) (e) V = Mat2,2 (R), U = {A E Mat2,2 (R) | trace(A) = 0}, V = Mat2,2 (R), U = {A E Mat2,2 (R) | det(A) = 0}, V = P3 (R), U = {fe P3 (R) | f(x²) = f(x)²}. Note: The trace of a matrix is the sum of its diagonal entries e.g. trace 3 = 1+4= 5.
Transcribed Image Text:1. In each of the following determine whether the set U is a subspace of the given vector space V. V = R², U = {(x, y) = R² | x+y=1}, V=R, U = Z, (b) (d) (e) V = Mat2,2 (R), U = {A E Mat2,2 (R) | trace(A) = 0}, V = Mat2,2 (R), U = {A E Mat2,2 (R) | det(A) = 0}, V = P3 (R), U = {fe P3 (R) | f(x²) = f(x)²}. Note: The trace of a matrix is the sum of its diagonal entries e.g. trace 3 = 1+4= 5.
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