Math 2568 Autumn 2020 Homework 6 Problem 1 Let L R4 R3 Be Gi ✓ Solved
Math 2568 Autumn 2020homework 6problem 1 Let L R4 R3 Be Given Byl
Given multiple mathematical problems involving linear transformations, vector spaces, and matrix operations, perform the following tasks for each respective problem:
1. For the linear transformation L : R4 → R3 defined by L([x1 x2 x3 x4])T) = [(x1 + 2x2 – x4) , (–2x1 + 3x2 + x3) , (x2 – 5x3 + 6x4)]T,
- Prove that L is a linear transformation.
- Find the matrix representation of L with respect to the standard bases of R4 and R3.
- Determine a basis for the kernel of L and compute its dimension.
- Determine a basis for the image of L and compute its dimension.
2. For the mappings from R3 to R2, analyze each and argue why it is or isn't a linear transformation:
- L([x1 x2 x3])T = [(x12 + 2x2), (–x1x3)]
- L([x1 x2 x3])T = [(3x1 – 2x2), (2x2 + x3 + 1)]
- L([x1 x2 x3])T = [(5x2 + 4x3), (x1 – 6x2 – x3)]
- L([x1 x2 x3])T = [(sin2(x1) + cos2(x1) + 7x2 – 1), (3x3 + e0 + sin(π))]
3. For the vector spaces of differentiable functions C1[a, b], evaluate whether each of the following maps is a linear transformation, providing justifications:
- L(f) = 3 ∫12 f(x) dx
- L(f) = f(5) + 2f(3) – f(1)
- L(f) = f(x) · sin(x) – 2f(x) ex
- L(f) = ∫24 f2(x) dx
4. For the matrix space M2×2(R), determine whether each of the following maps is a linear transformation, with supporting arguments:
- L(A) = det(A)
- L(A) = tr(A)
- L(A) = 2A – 3AT
- L(A) = 2A + A2
5. Given bases S1, S2, S3 for R3 and transition matrices between them, compute the following base transition matrices:
- S2 T S1
- S2 T S3
- S3 T S1
- S2 T S1 · S1 T S3 · S3 T S2
6. True or False: If a linear transformation L : R3 → R3 has rank 3, it is invertible.
7. True or False: If L1 : V → W and L2 : W → U are linear transformations, then their composition L2 · L1 : V → U is also linear.
8. True or False: For a linear transformation L : V → W with dim(V) = n and dim(W) = m, it holds that dim(ker(L)) + dim(im(L)) = m.