A. Is it a Subspace? Consider the following vector spaces V and subsets U. Determine if U is a subspace of V. Make sure to justify your findings. 1. U₁ = Set of 3 x 3 upper triangular matrices. V₁ = Set of all 3 x 3 matrices. (You may consider regular matrix addition and scalar multiplication.) 2. U₂ = Set of quadratic polynomials whose coefficients add up to 1. (examples: 3x²+2x −4; x²-x+1; 4x - 3) V₂ = P(2), set of all quadratic polynomials (ax² + bx + c) 3. U3 Set of 4 x 4 diagonal matrices. = V3 Set of all 4 x 4 matrices. =
A. Is it a Subspace? Consider the following vector spaces V and subsets U. Determine if U is a subspace of V. Make sure to justify your findings. 1. U₁ = Set of 3 x 3 upper triangular matrices. V₁ = Set of all 3 x 3 matrices. (You may consider regular matrix addition and scalar multiplication.) 2. U₂ = Set of quadratic polynomials whose coefficients add up to 1. (examples: 3x²+2x −4; x²-x+1; 4x - 3) V₂ = P(2), set of all quadratic polynomials (ax² + bx + c) 3. U3 Set of 4 x 4 diagonal matrices. = V3 Set of all 4 x 4 matrices. =
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 42EQ
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![A. Is it a Subspace?
Consider the following vector spaces V and subsets U. Determine if U is a subspace of V.
Make sure to justify your findings.
1. U₁ = Set of 3 x 3 upper triangular matrices.
V₁ = Set of all 3 x 3 matrices.
(You may consider regular matrix addition and scalar multiplication.)
2. U₂ = Set of quadratic polynomials whose coefficients add up to 1.
(examples: 3x²+2x − 4; x² − x + 1; 4x − 3)
V₂ = P(2), set of all quadratic polynomials (ax² + bx + c)
3. U3
V3
Set of 4 x 4 diagonal matrices.
Set of all 4 x 4 matrices.
B. Independently Linear
Which of the following sets of vectors are linearly independent? Explain your steps.
1. S₁ =
2. S₂ =
{][*]}
3. S3 = {(x² + 3x + 1), (x² − 2x), (3x² − x+1)}
C. All about that Basis
For each of the following sets of vectors S in a vector space V:
(i) Describe the subspace spanned by the set S.
(ii) Determine the dimension of Span(S).
(iii) Modify the set S to form a basis for V.
6
1. S₁ = {(1,0, -2, 1), (0, 1, 0, 1), (1, 2, -2,3)} in V₁ = R¹.
2. S₂ = {³-3x, 3x, x² - 2} in V₂ = P(3) (i.e., the space of polynomials of degree at
most 3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a081c38-6332-48ae-8325-0f0e5bc197cf%2F1ec824e4-500f-42e1-9a2b-f3e6a2852d29%2Fbjrb213_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A. Is it a Subspace?
Consider the following vector spaces V and subsets U. Determine if U is a subspace of V.
Make sure to justify your findings.
1. U₁ = Set of 3 x 3 upper triangular matrices.
V₁ = Set of all 3 x 3 matrices.
(You may consider regular matrix addition and scalar multiplication.)
2. U₂ = Set of quadratic polynomials whose coefficients add up to 1.
(examples: 3x²+2x − 4; x² − x + 1; 4x − 3)
V₂ = P(2), set of all quadratic polynomials (ax² + bx + c)
3. U3
V3
Set of 4 x 4 diagonal matrices.
Set of all 4 x 4 matrices.
B. Independently Linear
Which of the following sets of vectors are linearly independent? Explain your steps.
1. S₁ =
2. S₂ =
{][*]}
3. S3 = {(x² + 3x + 1), (x² − 2x), (3x² − x+1)}
C. All about that Basis
For each of the following sets of vectors S in a vector space V:
(i) Describe the subspace spanned by the set S.
(ii) Determine the dimension of Span(S).
(iii) Modify the set S to form a basis for V.
6
1. S₁ = {(1,0, -2, 1), (0, 1, 0, 1), (1, 2, -2,3)} in V₁ = R¹.
2. S₂ = {³-3x, 3x, x² - 2} in V₂ = P(3) (i.e., the space of polynomials of degree at
most 3)
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