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) 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}. (d) (e) Note: The trace of a matrix is the sum of its diagonal entries e.g. trace (²2) - 3 = 1+4= 5.
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) 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}. (d) (e) Note: The trace of a matrix is the sum of its diagonal entries e.g. trace (²2) - 3 = 1+4= 5.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
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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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I want to know the solution of question c,d,e, please.
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2Fd1d32001-ef74-4c8f-98c3-2bb6ff26092f%2F3iotqvs_processed.jpeg&w=3840&q=75)
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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