Transforming shapes with matrices

In this section, we will learn how to transform shapes with matrices. Instead of one column vector, we are going to have multiple vertices which create a shape. What we can do to this shape is use the transformation matrix to change the length and size of that shape. To do this computation, we merge all the vertices into one matrix and then multiply it with the transformation matrix. Doing so will give us another matrix. We will then take a look at each transformed vertices separately in the matrix to see the new transformed shape. Note that transforming the shape does not change the number of sides. We will take a look at some questions which involve transforming shapes, and then graph them to notice the changes between the normal shape and the transformed shape.

Let be vertices of a square and $T$ be a transformation matrix.
Then we can transform the square by combining the vertices into a matrix (denoted by $A$), and multiply it by the transformation matrix $T$. In other words,

And $TA$ is the transformed square.
Of course, this idea can also apply to other shapes other than squares.