NOTE: [a, b, c, d] <-- form is a vector andBOLDdenotes a vector, unless referring toR.

Suppose that T is a linear transformation fromR^4toR^3, defined by:

T(e_1) = [1, 5, 3], T(e_2) = [2, 6, 2], T(e_3) = [3, 7, 1], T(e_4) = [4, 8, 0].

1.) Find a standard Matrix A for the transformation.

2.) Find T([5, -1, 3, 2])

3.) Find anxsuch that T(x) = [-5, -9, 1]

4.) Is T 1-1? If yes, explain why it is. If not, give 2 different vectors that map to the same vector.

5.) Is T onto? If yes, explain why it is. If not, give a vector in the codomain that's not in the range.

WORK (PLEASE CHECK):

1.) The std. matrix A is:

[[1, 2, 3, 4], [5, 6, 7, 8], [3, 2, 1, 0]] (3x4 matrix)

2.) In order to find T([5, -1, 3, 2]), I would just take the matrix I formed in 1.) and multiply it by this vector, right?

3.) To find anxsuch that T(x) = [-5, -9, 1], I'd augment this vector to my matrix in 1.), row reduce, etc.

4.) NOT SURE!!

5.) It's clearly not onto, right? Since, we're taking a transformation fromR^4toR^3...everything can't possibly go from something bigger to something smaller.