Representing linear transformations with matrices

Now for the most common and important way of describing a linear transformation, the matrix. Through the magic of matrix-vector multiplication, a matrix is all you need to describe a linear transformation.

Again, let's start with an example. I'm going to describe the linear transformation we used in the An algebraic description section with a matrix. To jog your memory, here is the aforementioned linear transformation:

Now, here is how I can describe it with a matrix:

I don't even need to be that formal, other than telling you that we are using real numbers; I can just give you the matrix, and that describes everything. The dimension of the domain is the number of columns of the matrix, the dimension of the codomain is the number of rows of the matrix, and the actual transformation is the matrix itself. That is the power of a matrix!

Let's apply this transformation...