1. Scalar multiplication and vector addition are preserved, therefore this is a linear transformation.

2. Scalar multiplication and addition are preserved, therefore this is a linear transformation.

3.

A =

1 -3 5

0 1 -1

4.

a.

1 -2

1 0

b.

3 0

-1 2

5.

a. K(T) = [0, 0, x3]

b. T is not injective because the kernel includes nonzero vectors.

c. [x2, x1, 0]

d. T is not surjective because the range does not span the codomain.

e. T is not invertible.

Sorry for the poor formatting. Every row vector in problem 5 was originally a column vector. I just need to see if I'm going anywhere in the right direction here. Thanks.