I'm confused about the codomain of a linear transformation.
If we have a linear transformation which maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ and the range of the linear transformation is only the zero vector. Then what is the codomain? Is it $\mathbb{R}^m$ or the zero vector?
If you are given a linear transformation $T$ from $V$ to $W$, then the codomain is $W$. That's all, nothing to calculate, you just look it up.
If $W$ is not specified then the question has not been posed correctly and you cannot tell what the codomain is. (Though it may sometimes be implied by the context.)
The range is always a part of the codomain, but it may or may not be equal to the codomain.