In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis vectors. Picture of the grid.
These basis vectors $\hat i=<3,-2>$ and $\hat j=<2,1>$ can be put in a $2\times 2$ transformation matrix where they can apply the same transformation to any $\Bbb R^2$ vector.
$$T(\vec x)=\begin{bmatrix}3&2\\-2&1\end{bmatrix}\vec x$$
What kind of transformation matrices (besides one using a $2\times 2$ matrix) can be visualized by making a grid based off the column vectors. For example, is it possible to create a grid off the following transformation?
$$T(\vec x)=\begin{bmatrix}3&2&2\\-2&1&0\end{bmatrix}\vec x$$
Or is creating a grid only possible if I know that the columns of the matrix correspond to basis vectors?