So, we’ve jumped around a bit in what is turning into an introduction to linear algebra. The posts here, here, here, here, here, and here show the ground we’ve covered so far—although, saying it that way implies that we’ve moved along continuous patches of ground, which is certainly not true. We skipped over adding and scaling vectors and have focused on concepts which have close analogs to current high school algebra and geometry topics.

Now we’ll jump to the concept of a matrix. A matrix gives you information about two arrows—the x-axis arrow, if you will, and the y-axis arrow. The matrix below, for example, tells you that you are in the familiar xy coordinate plane, with the x arrow, or x vector, extending from the origin to (1, 0) and the y arrow, or y vector, going from the origin to (0, 1).

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

This is a kind of home-base matrix, and it is called the identity matrix. If we multiply a vector by this matrix, we’ll always get back the vector we put in. The equation below shows how this matrix-vector multiplication is done with the identity matrix and the vector (1, 2), as shown at the right.

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

As you can see on the far right of the equation, the result is (1 + 0, 0 + 2), or (1, 2), the vector we started with.

A Linear Transformation

Now let’s take the vector at (1, 2) and map it to (0, 2). We’re looking for a matrix that can accomplish this—a transformation of the coordinate system that will map (1, 2) to (0, 2). If we shrink the horizontal vector to (0, 0) and keep the vertical vector the same, that would seem to do the trick.

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

And it does! This matrix is called a shear matrix, and it takes any vector and shmooshes it onto the y-axis. We could do the same for any vector and the x-axis by zeroing out the second column of the matrix and keeping the first column the same.

You can try out all kinds of different numbers to see their effects. You can do rotations, reflections, and scalings, among other things. The transformation shown at right, for example, where the two column vectors are taken to (1, 1) and (–1, 1), respectively, maps the vector (1, 2) to the vector (–1, 3).

\(\begin{bmatrix}\mathtt{1} & \mathtt{-1}\\\mathtt{1} & \mathtt{\,\,\,\,1}\end{bmatrix}\begin{bmatrix}\mathtt{1}\\\mathtt{2}\end{bmatrix} = \mathtt{1}\begin{bmatrix}\mathtt{1}\\\mathtt{1}\end{bmatrix} + \mathtt{2}\begin{bmatrix}\mathtt{-1}\\\mathtt{\,\,\,\,1}\end{bmatrix} = \begin{bmatrix}\mathtt{(1)(1) + (2)(-1)}\\\mathtt{(1)(1) + (2)(1)}\end{bmatrix}\)

You may notice, by the way, that what we did with the matrix above was to first rotate the column vectors by 45° and then scale them up by a factor of \(\mathtt{\sqrt{2}}\). We can do each of these transformations with just one matrix. \[\begin{bmatrix}\mathtt{\frac{\sqrt{2}}{\,\,2}} & \mathtt{\frac{-\sqrt{2}}{\,\,2}}\\\mathtt{\frac{\sqrt{2}}{\,\,2}} & \mathtt{\,\,\,\,\frac{\sqrt{2}}{2}}\end{bmatrix} \leftarrow \textrm{Rotate by 45}^\circ \textrm{.} \quad \quad \begin{bmatrix}\mathtt{\sqrt{2}} & \mathtt{0}\\\mathtt{0} & \mathtt{\sqrt{2}}\end{bmatrix} \leftarrow \textrm{Scale up by }\sqrt{2}\textrm{.}\]

Then, we can combine these matrices by multiplying them to produce the transformation matrix we needed. Each column of one of the matrices is multiplied by both columns of the other to get the two column vectors of the resulting matrix. We’ll look at that more in the future.