In this section, we describe some specific linear transformations in [math]\mathbb{R}^2[/math] that are geometric in nature. They are as follows:[br][br][list=1][*]Rotation about the origin through a given angle[/*][*]Reflection in a line that passes through the origin[br][/*][*]Scaling horizontally or vertically[/*][*]Horizontal or vertical shear[/*][*]Projection onto x-axis or y-axis[/*][/list][br]In the applet, you can see the effect on [math]\Delta CDE[/math] under various geometric transformations: First, you select the type of geometric linear transformation [math]T[/math]. Second, you click the "Set" button to set up the corresponding [math]T(\hat{\mathbf{i}})[/math] and [math]T(\hat{\mathbf{j}})[/math] and the matrix for [math]T[/math]. Then click the "Go" button to see how the triangle is transformed under [math]T[/math]. [br]
What is the general form of the matrix for the rotation about the origin through angle [math]\alpha[/math] in the counterclockwise direction?
Find the matrix for the 2D linear transformation that first rotates about the origin through [math]45^\circ[/math] in the counterclockwise direction and then reflect on the y-axis.[br][br]