The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Representing transformations with matrices

The transformations discussed on the previous page are conveniently expressed using vector notation. The usual convention is to write the coordinates of a point in a vertical column (rather than a horizontal pair) and to use square brackets. With this notation, the equations for a translation become

so that translation is just seen to be adding a constant. A scale factor can be similarly expressed as multiplication by a constant (or scalar)


A rotation is a bit more complicated. The general form for a rotation counter-clockwise through an angle α is given as follows:

(See a derivation of these equations here).

Using matrix notation, we can separate out the variables x and y from their coefficients, writing


Here, the coefficients have been arranged in a table with two rows and two columns, namely . This is called a 2 by 2 matrix. The product of a matrix with a column vector is defined as in the equations above. The top entry of the result is ax + by and the second entry is cx + dy. Forming these results is sometimes referred to as multiplying the column vector by the first and second rows of the matrix, respectively.

Returning to the case of rotation, note that for any particular angle, the cosine and sine terms will be constants. For example, we might choose the angle α so that the cosine is 3/5 and the sine is 4/5. (This gives one of the acute angles in a 3-4-5 right triangle.) To rotate through this angle α, our equation becomes

Similarly, to rotate through π/2, since we know that cos π/2 = 0 and sin π/2 = 1, we find

which agrees with the example on the previous page.

Although we have already seen that scale factor transformations amount to multiplication by a fixed constant, we can also express these transformations in matrix form. The equation for a scale factor of r is


As a general matter, any 2 by 2 matrix can be used to define a transformation, and every such transformation can be understood geometrically. So far, we have seen matrix formulations for rotations and scale factors. We will look at a couple more specific kinds of transformations next.