How to use a transformation matrix to map points and figures
A transformation matrix tells us where each point moves under a linear transformation. We place the point’s coordinates in a column and multiply. Applying the same matrix to a figure’s vertices shows how the whole figure changes.
Transform a single point
For the point , multiply its coordinate column by .
The point moves to . The same matrix gives a new point for every input in the plane.
Put a figure’s vertices into columns
For a triangle with vertices , , and , use the following transformation and vertex matrix.
| Original | Transformed |
|---|---|
Read the columns as images of unit vectors
The first column of tells us where goes, so . The second tells us where goes, so . Every vector is a combination of those two unit directions, which is why the two columns determine the entire linear transformation.
Build a matrix from those two images
If maps to and maps to , use those images as the columns.
This matrix sends to . A two-by-two matrix always sends the origin to the origin. A translation is represented by an affine rule with an added vector, or by a larger homogeneous-coordinate matrix.