Suppose we place two points and in the plane. They lie on the and axes, respectively, at one unit of distance from the origin . Using vector spaces, these two points would be the standard unit basis vectors (sometimes written as and ). Watch what happens if we substitute them into (3.5):

and

These special points simply select the column vectors on . What does this mean? If is applied to transform a model, then each column of indicates precisely how each coordinate axis is changed.

Figure 3.5 illustrates the effect of applying various matrices to a model. Starting with the upper right, the identity matrix does not cause the coordinates to change:
. The second example causes a flip as if a mirror were placed at the axis. In this case,
. The second row shows examples of scaling. The matrix on the left produces
, which doubles the size. The matrix on the right only stretches the model in the direction, causing an *aspect ratio* distortion. In the third row, it might seem that the matrix on the left produces a mirror image with respect to both and axes. This is true, except that the mirror image of a mirror image restores the original. Thus, this corresponds to the case of a -degree ( radians) rotation, rather than a mirror image. The matrix on the right produces a shear along the direction:
. The amount of displacement is proportional to . In the bottom row, the matrix on the left shows a skew in the direction. The final matrix might at first appear to cause more skewing, but it is degenerate. The two-dimensional shape collapses into a single dimension when is applied:
. This corresponds to the case of a *singular* matrix, which means that its columns are not linearly independent (they are in fact identical). A matrix is singular if and only if its determinant is zero.

Steven M LaValle 2020-11-11