I need to find the dimension of the image of the linear transformation $f(v)$, where $f\colon \Bbb R^2 \to \Bbb R^2$ is defined by
$$f(v) = \begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}v$$
I have already found that the linear transformation is neither injective nor surjective by finding contradictory examples for both, but I'm not sure where to proceed. I know that the $\text{Im} (f) = \{f(v) \} v$ elements of vector space $V$}, but I'm not sure how to derive the dimension.
Best Answer
$\dim(f)=\dim([1,0;0,0])$ because matrix of $f$ in standard basis is exactly $[1,0;0,0]$