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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?

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    $\begingroup$ This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$. $\endgroup$ – tch Jan 16 at 19:50
  • $\begingroup$ How do I visualize a grid for the transformation matrix with the column $<2,0>$? $\endgroup$ – mrhumanzee Jan 16 at 20:11
  • $\begingroup$ You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $\mathbb{R}^3$ to $\mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $\mathbb{R}^3$ to a 2d plane. $\endgroup$ – tch Jan 16 at 20:51

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