Linear algebra questions

Math 2568 Autumn 2020

Homework 6

Problem 1 Let L : R4 → R3 be given by

L ([ x1 x2 x3 x4

]T) = [ (x1 + 2x2 − x4) (−2x1 + 3x2 + x3) (x2 − 5x3 + 6x4)

]T • Show that L is a linear transformation.

• Find the matrix representation of L with respect to the standard bases of R4 and R3.

• Find a basis for ker(L) and compute its dimension.

• Find a basis for im(L) and compute its dimension.

Problem 2 Determine which of the following maps are linear transformations from R3 → R2. In each give an argument showing it is, or a reason explaining why it is not.

• L ([ x1 x2 x3

]T) = [ (x21 + 2x2) (−x1x3)

]T • L

([ x1 x2 x3

]T) = [ (3x1 − 2x2) (2x2 + x3 + 1)

]T • L

([ x1 x2 x3

]T) = [ (5x2 + 4x3) (x1 − 6x2 − x3)

]T • L

([ x1 x2 x3

]T) = [ (sin2(x1) + cos

2(x1) + 7x2 − 1) (3x3 + e0 + sin(π)) ]T

Problem 3 Let C1[a, b] denote the vector space of real-valued differentiable functions on the interval [a, b]. Determine which of the following maps are linear transformations. In each give an argument showing it is, or a reason explaining why it is not.

• L : C1[1, 2]→ R;L(f) = 3 ∫ 2 1 f(x) dx

• L : C1[1, 7]→ R;L(f) = f(5) + 2f(3)− f(1)

• L : C1[3, 5]→ C1[3, 5];L(f)(x) = f(x) · sin(x)− 2f(x)ex

• L : C1[2, 4]→ R;L(f) = ∫ 4 2 f2(x) dx

Problem 4 Let M2×2(R) denote the vector space of real 2×2 matrices. Determine which of the following maps are linear transformations. In each give an argument showing it is, or a reason explaining why it is not.

• L : M2×2(R)→ R;L(A) = Det(A)

• L : M2×2(R)→ R;L(A) = Tr(A) (here Tr(A) = A(1, 1) +A(2, 2) is the trace of A)

• L : M2×2(R)→ L : M2×2(R);L(A) = 2 ∗A− 3 ∗AT

• L : M2×2(R)→ L : M2×2(R);L(A) = 2 ∗A+A2

Problem 5 Suppose S1, S2, S3 are three bases for R3 with S1TS2 =

2 3 −11 0 −2 0 −3 5

 and S1TS3 = 0 −3 14 6 0

8 0 −3

. Compute the following base transition matrices:

• S2TS1

• S2TS3

• S3TS1

• S2TS1 ∗ S1TS3 ∗ S3TS2

For the remaining two problems, you are to determine of if the given statement is true or false, and provide a short justification for your answer (for example, an argument showing it is true, or a counter-example if it is false).

Problem 6 If a linear transformation L : R3 → R3 has rank 3, it is invertible.

Problem 7 If L1 : V →W and L2 : W → U are linear transformations, then so is L2 ◦ L1 : V → U .

Problem 8 If L : V →W is a linear transformation with dim(V ) = n, dim(W ) = m, then dim(ker(L))+ dim(im(L)) = m.