Define what is meant by the terms rotation, reflection, dilation and shear. Give examples of real $2 \times 2$ matrices representing each of these.

Consider the three $2 \times 2$ matrices

$$A=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right), \quad B=\frac{1}{\sqrt{2}}\left(\begin{array}{ll} 1 & 1 \\ 1 & 3 \end{array}\right) \quad \text { and } \quad C=A B$$

Identify the three matrices in terms of your definitions above.