Proof that the dimension of the image of a transformation is the rank of the matrix that induces this transformation.

Question

If $T$ is a transformation, we can write $T(v)=Av$, where $A$ is the matrix that induces that transformation on $v$.

We were asked for a rigorous proof of the question I mentioned above, and I was completely blank. I always assumed it was a de facto thing.

How do I go about proving this?

Answer 1

Hint:

For a square matrix $A$, the rank of $A$ is the rank of its column vectors, and these generate the image of the associated linear map.