Confused about dimension of a range of a linear map Suppose you have a linear map $T:Mat_{2\times3}(R)\rightarrow Mat_{2\times2}(R)$
Why is the range(T) 4 at most? I thought that this is a 2x2 matrix and that the maximum range is 2?

Range$(T)$ is a subspace of $M_{2\times 2}(R)$, which is a vector space of dimension $4$. Because the dimension of any subspace is less than or equal to its parent space, it follows that the largest possible dimension of Range$(T)$ is $4$.
You can verify that the dimension of $M_{2\times 2}(R)$ is $4$ by verifying that the list of matrices $$\begin{bmatrix}1 & 0 \\ 0& 0 \end{bmatrix}, \begin{bmatrix}0 & 1 \\ 0& 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \\ 1& 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0& 1 \end{bmatrix} $$ is linearly independent and spans $M_{2\times 2}(R)$.
