Problem 5 (Rank-Nullity Theorem). Let T : P3 → M2×2 be defined as, T(p(x)) P(0) p'(1)] = 1. Prove that T is a linear transformation. 2. Find ker(T). Is T injective? 3. Find im(T). Is T surjective? 4. Verify the Rank-Nullity Theorem for T. Problem 6 (Change of Basis). Let B₁ = polynomials in P3. - - {1, x, x², x³} and B₁ = {1, x, x(x − 1), x(x − 1)(x − 2)} be two sets of 1. Is B2 a basis for P3? Justify your answer. 2. Find SB₁→B₂ and SB2→B₁. Which one is "easier" to find? - Problem 7 (Change of Basis). Let B₁ = {eª, sin² (x), cos² (x)} and B₁ = {e*, sin(2x)}. Recall that sin(20) = 2 sin(0) cos(0). Suppose V = span (B₁) and W = span(B2). Let T: VW be a linear transformation defined as T(f(x)) = f'(x). 1 1. Prove that B₁ is a basis. 2. Let g(x) = 5 - 3e. Show that g = V and find T(g(x)). 3. Find [TB₁B2 4. Is T injective? 5. Is T surjective?
Problem 5 (Rank-Nullity Theorem). Let T : P3 → M2×2 be defined as, T(p(x)) P(0) p'(1)] = 1. Prove that T is a linear transformation. 2. Find ker(T). Is T injective? 3. Find im(T). Is T surjective? 4. Verify the Rank-Nullity Theorem for T. Problem 6 (Change of Basis). Let B₁ = polynomials in P3. - - {1, x, x², x³} and B₁ = {1, x, x(x − 1), x(x − 1)(x − 2)} be two sets of 1. Is B2 a basis for P3? Justify your answer. 2. Find SB₁→B₂ and SB2→B₁. Which one is "easier" to find? - Problem 7 (Change of Basis). Let B₁ = {eª, sin² (x), cos² (x)} and B₁ = {e*, sin(2x)}. Recall that sin(20) = 2 sin(0) cos(0). Suppose V = span (B₁) and W = span(B2). Let T: VW be a linear transformation defined as T(f(x)) = f'(x). 1 1. Prove that B₁ is a basis. 2. Let g(x) = 5 - 3e. Show that g = V and find T(g(x)). 3. Find [TB₁B2 4. Is T injective? 5. Is T surjective?
Elementary Linear Algebra (MindTap Course List)
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Chapter7: Eigenvalues And Eigenvectors
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Problem 25CM: Find a basis B for R3 such that the matrix for the linear transformation T:R3R3,...
Related questions
Question
![Problem 5 (Rank-Nullity Theorem). Let T : P3 → M2×2 be defined as,
T(p(x))
P(0) p'(1)]
=
1. Prove that T is a linear transformation.
2. Find ker(T). Is T injective?
3. Find im(T). Is T surjective?
4. Verify the Rank-Nullity Theorem for T.
Problem 6 (Change of Basis). Let B₁ =
polynomials in P3.
-
-
{1, x, x², x³} and B₁ = {1, x, x(x − 1), x(x − 1)(x − 2)} be two sets of
1. Is B2 a basis for P3? Justify your answer.
2. Find SB₁→B₂ and SB2→B₁. Which one is "easier" to find?
-
Problem 7 (Change of Basis). Let B₁ = {eª, sin² (x), cos² (x)} and B₁ = {e*, sin(2x)}. Recall that sin(20) =
2 sin(0) cos(0). Suppose V = span (B₁) and W = span(B2). Let T: VW be a linear transformation defined
as
T(f(x)) = f'(x).
1
1. Prove that B₁ is a basis.
2. Let g(x) = 5 - 3e. Show that g = V and find T(g(x)).
3. Find [TB₁B2
4. Is T injective?
5. Is T surjective?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7a0dc4c-c17f-401f-9f65-9d6c8cbbca58%2F3adbd89a-d9e0-4533-8b75-3009f939ae64%2Fmsskzzk_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 5 (Rank-Nullity Theorem). Let T : P3 → M2×2 be defined as,
T(p(x))
P(0) p'(1)]
=
1. Prove that T is a linear transformation.
2. Find ker(T). Is T injective?
3. Find im(T). Is T surjective?
4. Verify the Rank-Nullity Theorem for T.
Problem 6 (Change of Basis). Let B₁ =
polynomials in P3.
-
-
{1, x, x², x³} and B₁ = {1, x, x(x − 1), x(x − 1)(x − 2)} be two sets of
1. Is B2 a basis for P3? Justify your answer.
2. Find SB₁→B₂ and SB2→B₁. Which one is "easier" to find?
-
Problem 7 (Change of Basis). Let B₁ = {eª, sin² (x), cos² (x)} and B₁ = {e*, sin(2x)}. Recall that sin(20) =
2 sin(0) cos(0). Suppose V = span (B₁) and W = span(B2). Let T: VW be a linear transformation defined
as
T(f(x)) = f'(x).
1
1. Prove that B₁ is a basis.
2. Let g(x) = 5 - 3e. Show that g = V and find T(g(x)).
3. Find [TB₁B2
4. Is T injective?
5. Is T surjective?
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