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.
1 -3 5
0 1 -1
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.