Matrix representation for a transformation

Homework Statement

Consider the linear transformation T: R2-->R2, where
T(x1, x2, x3)= (3x2-x3, x1+4x2+x3)

a. Find a matrix which implements this mapping.
b. Is this transformation one-to-one? Is it onto? Explain.

Homework Equations

[T(x)]_B = ([T]_B) (x_B)

The Attempt at a Solution

The matrix that implements this mapping would be the representation ([T]_B). I think that (x_B) is the vector [x1, x2, x3] and that [T(x)]_B is (3x2-x3, x1+4x2+x3) relative to the {x1, x2, x3} basis. So then ([T]_B) must be the matrix:

0 3 -1
1 4 1

Row-reducing this matrix to echelon form gives 2 pivots, so the transformation is onto since the system is consistent, but it is not one-to-one because the system is linearly dependent.

Are all my thoughts correct for this problem?