I'm just trying to understand the difference between questions - if something says "find the image of the matrix" vs. "find the basis of the image of the matrix" is there any difference?
And if so, when given a matrix, what is the difference between the two?
Thank you!
The image $\text{Im}(A)$ of a matrix $A\in\mathbb{M}_{n\times m}(K)$ is a vector subspace of $K^n$, since for vectors $u,v\in K^m$ and scalars $\lambda,\mu\in K$ we have $$\lambda Av+\mu Aw=A(\lambda v+\mu w).$$
Therefore, when you are told to show $\text{Im}(A)$, a good way to do so is, as with any other vector subspace, to show a basis for it.