A practice problem I have asks us to find the rank of the linear transformation:
$$T(x, y, z) = (x - 2y + z, 2x + y + z)$$
What I'm confused on is, thus far in my class, as far as I can recall, we've only ever found the rank of linear transformations in the form $T(x) = Ax$, where you have a matrix $--$ and you put the matrix in reduced row-echelon form and see how many leading $1$'s there are in order to determine the rank. So since we don't have a matrix here, I'm not sure how I would solve this.
I'm getting the sense that it's possible to put ANY linear transformation into matrix form, but the problem is my book hasn't reached that point yet; I'm pretty sure that's in the next section. So I don't know how to represent this transformation with a matrix (and I don't think I'm supposed to since we haven't learned that yet), but I don't know how I would solve this problem without doing that.
The problem also says NOT to compute the kernel, so I can't find the kernel and then use the rank-nullity theorem to find the rank, either.
I'm at all loss as to how I would do this. I've flipped through the section and all the problems in my book and I don't see any examples that compute the rank WITHOUT having a matrix.
Is there another way to compute the rank of this that I'm missing?