Okay, now let’s move stuff around with linear algebra. We’ll eventually do rotations, reflections, and maybe translations too, while mixing that up with stretchings and skewings and other things that matrices can do for us.

We learned here that a matrix gives us information about two arrows—the x-axis arrow and the y-axis arrow. What we really mean is that a 2 × 2 matrix represents a transformation of 2D space. This transformation is given by 2 column vectors—the 2 columns of the matrix. The identity matrix, as we saw previously, represents the do-nothing transformation:

\[\begin{bmatrix}\mathtt{\color{blue}{1}} & \mathtt{\color{orange}{0}}\\\mathtt{\color{blue}{0}} & \mathtt{\color{orange}{1}}\end{bmatrix} \leftarrow \begin{bmatrix}\mathtt{\color{blue}{1}}\\\mathtt{\color{blue}{0}}\end{bmatrix} \text{and} \begin{bmatrix}\mathtt{\color{orange}{0}}\\\mathtt{\color{orange}{1}}\end{bmatrix}\]

Another way to look at this matrix is that it tells us about the 2D space we’re looking at and how to interpret ANY vector in that space. So, what does the vector (1, 2) mean here? It means take 1 of the (1, 0) vectors and add 2 of the (0, 1) vectors.

\[\begin{bmatrix}\mathtt{1} & \mathtt{0}\\\mathtt{0} & \mathtt{1}\end{bmatrix}\begin{bmatrix}\mathtt{1}\\\mathtt{2}\end{bmatrix} = \mathtt{1}\begin{bmatrix}\mathtt{1}\\\mathtt{0}\end{bmatrix} + \mathtt{2}\begin{bmatrix}\mathtt{0}\\\mathtt{1}\end{bmatrix} = \begin{bmatrix}\mathtt{(1)(1) + (2)(0)}\\\mathtt{(1)(0) + (2)(1)}\end{bmatrix}\]

But what if we reflect the entire coordinate plane across the y-axis? That’s a new system, and it’s a system given by where the blue and orange vectors would be under that reflection:

\[\begin{bmatrix}\mathtt{\color{blue}{-1}} & \mathtt{\color{orange}{0}}\\\mathtt{\color{blue}{\,\,\,\,0}} & \mathtt{\color{orange}{1}}\end{bmatrix} \leftarrow \begin{bmatrix}\mathtt{\color{blue}{-1}}\\\mathtt{\color{blue}{\,\,\,\,0}}\end{bmatrix} \text{and} \begin{bmatrix}\mathtt{\color{orange}{0}}\\\mathtt{\color{orange}{1}}\end{bmatrix}\]

In that new system, we can guess where the vector (1, 2) will end up. It will just be reflected across the y-axis. But matrix-vector multiplication allows us to figure that out by just multiplying the vector and the matrix:

\[\begin{bmatrix}\mathtt{-1} & \mathtt{0}\\\mathtt{\,\,\,\,0} & \mathtt{1}\end{bmatrix}\begin{bmatrix}\mathtt{1}\\\mathtt{2}\end{bmatrix} = \mathtt{1}\begin{bmatrix}\mathtt{-1}\\\mathtt{\,\,\,\,0}\end{bmatrix} + \mathtt{2}\begin{bmatrix}\mathtt{0}\\\mathtt{1}\end{bmatrix} = \begin{bmatrix}\mathtt{-1}\\\mathtt{\,\,\,\,2}\end{bmatrix}\]

This opens up a ton of possibilities for specifying different kinds of transformations. And it makes it pretty straightforward to specify transformations and play with them—just set the two column vectors of your matrix and see what happens! We can rotate and reflect the column vectors and scale them up together or separately.

Rotations

Let’s start with rotations. And we’ll throw in some scaling too, just to make it more interesting. The image shows a coordinate system that has been rotated –135°, by rotating our column vectors from the identity matrix by that degree. The coordinate system has also been dilated by a factor of 0.5. This results in \(\mathtt{\triangle{ABC}}\) rotated –135° and scaled down by a half as shown.

What matrix represents this new rotated and scaled down system? The rotation of the first column vector, (1, 0), can be represented as (\(\mathtt{cos\,θ, sin\,θ}\)). And the second column vector, which is (0, 1) before the rotation, is perpendicular to the first column vector, so we just flip the components and make one of them the opposite of what it originally was:

(\(\mathtt{-sin\,θ, cos\,θ}\)). So, a general rotation matrix looks like the matrix on the left. The rotation matrix for a –135° rotation is on the right: \[\begin{bmatrix}\mathtt{cos \,θ} & \mathtt{-sin\,θ}\\\mathtt{sin\,θ} & \mathtt{\,\,\,\,\,cos\,θ}\end{bmatrix}\quad\quad\begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{2}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{2}}\\\mathtt{-\frac{\sqrt{2}}{2}} & \mathtt{-\frac{\sqrt{2}}{2}}\end{bmatrix}\]

You can eyeball that the rotation matrix is correct by interpreting the columns of the matrix as the new positions of the horizontal vector and vertical vector, respectively (the new coordinates they are pointing to). A –135° rotation is a clockwise rotation of 90° + 45°.

Now for the scaling, or dilation by a factor of 0.5. This is accomplished by the matrix on the left, which, when multiplied by the rotation matrix on the right, will give us the one combo transformation matrix: \[\begin{bmatrix}\mathtt{\frac{1}{2}} & \mathtt{0}\\\mathtt{0} & \mathtt{\frac{1}{2}}\end{bmatrix}\begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{2}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{2}}\\\mathtt{-\frac{\sqrt{2}}{2}} & \mathtt{-\frac{\sqrt{2}}{2}}\end{bmatrix} = \begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{4}}\\\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{-\frac{\sqrt{2}}{4}}\end{bmatrix}\]

The result is another 2 × 2 matrix, with two column vectors. The calculations below show how we find those two new column vectors: \[\mathtt{-\frac{\sqrt{2}}{2}}\begin{bmatrix}\mathtt{\frac{1}{2}}\\\mathtt{0}\end{bmatrix} + -\frac{\sqrt{2}}{2}\begin{bmatrix}\mathtt{0}\\\mathtt{\frac{1}{2}}\end{bmatrix} = \begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{4}}\\\mathtt{-\frac{\sqrt{2}}{4}}\end{bmatrix}\quad\quad\mathtt{\frac{\sqrt{2}}{2}}\begin{bmatrix}\mathtt{\frac{1}{2}}\\\mathtt{0}\end{bmatrix} + -\frac{\sqrt{2}}{2}\begin{bmatrix}\mathtt{0}\\\mathtt{\frac{1}{2}}\end{bmatrix} = \begin{bmatrix}\mathtt{\,\,\,\,\frac{\sqrt{2}}{4}}\\\mathtt{-\frac{\sqrt{2}}{4}}\end{bmatrix}\]

Now for the Point of Rotation

We’ve got just one problem left. Our transformation matrix, let’s call it \(\mathtt{A}\), is perfect, but we don’t rotate around the origin. So, we have to do some adding to get our final expression. To rotate, for example, point B around point C, we don’t use point B’s position vector from the origin—we rewrite this vector as though point C were the origin. So, point B has a position vector of B – C = (1, 0) in the point C–centered system. Once we’re done rotating this new position vector for point B, we have to add the position vector for C back to the result. So, we get: \[\mathtt{B’} = \begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{4}}\\\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{-\frac{\sqrt{2}}{4}}\end{bmatrix}\begin{bmatrix}\mathtt{1}\\\mathtt{0}\end{bmatrix} + \begin{bmatrix}\mathtt{2}\\\mathtt{2}\end{bmatrix} = \begin{bmatrix}\mathtt{2\,-\,\frac{\sqrt{2}}{4}}\\\mathtt{2\,-\,\frac{\sqrt{2}}{4}}\end{bmatrix}\]

Which gives us a result, for point B’, of approximately (1.65, 1.65). We can do the calculation for point A as well: \[\,\,\,\,\,\mathtt{A’} = \begin{bmatrix}\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{4}}\\\mathtt{-\frac{\sqrt{2}}{4}} & \mathtt{-\frac{\sqrt{2}}{4}}\end{bmatrix}\begin{bmatrix}\mathtt{-1}\\\mathtt{\,\,\,\,2}\end{bmatrix} + \begin{bmatrix}\mathtt{2}\\\mathtt{2}\end{bmatrix} = \begin{bmatrix}\mathtt{2\,+\,\frac{3\sqrt{2}}{4}}\\\mathtt{2\,-\,\frac{\sqrt{2}}{4}}\end{bmatrix}\]

This puts A’ at about (3.06, 1.65). Looks right! By the way, the determinant is \(\mathtt{\frac{1}{4}}\)—go calculate that for yourself. This is no surprise, of course, since a dilation by a factor of 0.5 will scale areas down by one fourth. The rotation has no effect on the determinant, because rotations do not affect areas.

Our general formula, then, for a rotation through \(\mathtt{θ}\) of some point \(\mathtt{x}\) (as represented by a position vector) about some point \(\mathtt{r}\) (also represented by a position vector) is: \[\mathtt{x’} = \begin{bmatrix}\mathtt{cos\,θ} & \mathtt{-sin\,θ}\\\mathtt{sin\,θ} & \mathtt{\,\,\,\,\,cos\,θ}\end{bmatrix}\begin{bmatrix}\mathtt{x_1\,-\,r_1}\\\mathtt{x_2\,-\,r_2}\end{bmatrix} + \begin{bmatrix}\mathtt{r_1}\\\mathtt{r_2}\end{bmatrix}\]