Example2.6.3

In this example, we will find the matrix defining a matrix transformation that performs a $$45^\circ$$ counterclockwise rotation.

We first need to know that this geometric operation can be represented by a matrix transformation. To begin, we will define the function $$T:\real^2\to\real^2$$ where $$T(\xvec)$$ is obtained by rotating $$\xvec$$ counterclockwise by $$45^\circ\text{,}$$ as shown in Figure 2.6.4.

We need to check that $$T$$ is a matrix transformation; by Proposition 2.6.2, this means that we should make sure that

\begin{equation*} \begin{aligned} T(c\vvec) \amp {}={} cT(\vvec) \\ T(\vvec + \wvec) \amp {}={} T(\vvec) + T(\wvec)\text{.} \end{aligned} \end{equation*}

The next two figures illustrate these properties. For instance, Figure 2.6.5 shows that relationship between $$T(\vvec)$$ and $$T(c\vvec)$$ when $$c$$ is a scalar. We easily see that $$T(c\vvec)$$ is a scalar multiple of $$T(\vvec)$$ and hence that $$T(c\vvec) = cT(\vvec)\text{.}$$

Similarly, Figure 2.6.6 shows the relationship between $$T(\vvec+\wvec)\text{,}$$ $$T(\vvec)\text{,}$$ and $$T(\wvec)\text{.}$$ In this way, we see that $$T(\vvec+\wvec) = T(\vvec) + T(\wvec)\text{.}$$

This shows that the function $$T\text{,}$$ which rotates vectors by $$45^\circ$$ is a matrix transformation. We may therefore write it as $$T(\xvec) = A\xvec$$ where $$A$$ is the $$2\times2$$ matrix $$A=\left[\begin{array}{rr} T(\evec_1) \amp T(\evec_2) \end{array}\right]\text{.}$$ The columns of this matrix, $$T(\evec_1)$$ and $$T(\evec_2)\text{,}$$ are shown in Figure 2.6.7.

To find the components of these vectors, notice that they form an isosceles right triangle, as shown in Figure 2.6.8. Since the length of $$\evec_1$$ is 1, the length of $$T(\evec_1)\text{,}$$ the hypotenuse of the triangle, is 1.

Hence, the matrix $$A$$ is
You may wish to check this using the interactive diagram in the previous activity using the approximation $$1/\sqrt{2} \approx 0.7\text{.}$$