This applet shows the effect of a linear transformation [math]T:\mathbb{R}^2 \rightarrow \mathbb{R}^2[/math].[br][br]The effects of [math]T[/math] on the blue vector [color=#0000FF][math]\vec{v}[/math][/color] and the [color=#0000FF]blue triangle[/color] are depicted as the red vector [color=#FF0000][math]T(\vec{v})[/math][/color] and the [color=#FF0000]red triangle[/color].[br][br]The matrix corresponding to [math]T[/math] is called [math]A[/math]. Recall that the column vectors of [math]A[/math] are given by [math]T(\vec{e}_1)[/math] and [math]T(\vec{e}_2)[/math], where [math]\vec{e}_1=\begin{pmatrix}1\\0\end{pmatrix}[/math] and [math]\vec{e}_2=\begin{pmatrix}0\\1\end{pmatrix}[/math].[br][br]Manipulate [math]T(\vec{e}_1)[/math] and [math]T(\vec{e}_2)[/math] to define a new linear transformation and see its corresponding matrix. You may also manipluate [color=#0000FF][math]\vec{v}[/math][/color] and the [color=#0000FF]blue triangle[/color].

[b]Exercises[/b][br][br]For the following four exercises, find the matrix for the linear transformation corresponding to[br][list=5][br][*] scaling by the factor 1/2.[br][*] reflection across the line [math]y=-x[/math].[br][*] 180 degree rotation about the origin.[br][*] projection onto the [math]y[/math]-axis.[br][/list][br][br]For the next four exercises, describe the linear transformation given by the matrix[br][list=5][br][*] [math]\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}[/math][br][*] [math]\begin{bmatrix}0.5 \quad 0.5\\0.5 \quad 0.5\end{bmatrix}[/math][br][*] [math]\begin{bmatrix}0 \quad 1\\1 \quad 0\end{bmatrix}[/math][br][*] [math]\begin{bmatrix}0 \quad 2\\2 \quad 0\end{bmatrix}[/math][br][/list]