###### Example 2.6.3

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

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.

This leads to

Hence, the matrix \(A\) is

You may wish to check this using the interactive diagram in the previous activity using the approximation \(1/\sqrt{2} \approx 0.7\text{.}\)