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4. Let R be the rotation of a vector by 45 degrees (?/4 radians) about the origin. Find the equation of the line produced by rotating the line y = -2x using the transformation R. You may take for granted that a linear transformation transforms a line into another line. 5. In this problem, the transformations R, D, and T are all linear transformations from R2 to R2. You do not need to verify their linearity. (a) Let R be the transformation that reflects elements of R2 across the line y = x. Let D be the transformation that scales the vertical direction by 2/3 and scales the horizontal direction by 3/2. Find the matrix representations of R and D. (b) Let T be given by $T\begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} -3 \ 3/2 \end{pmatrix}$ and $T\begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} -4 \ 3 \end{pmatrix}$. Find the matrix representation of T using the structure theorem. (c) The transformation T can be decomposed into basic transformations. Namely, it is the dilation D, followed by a shear S, and lastly the reflection R. Find the matrix for the shear S. The transformations R and D are those from part (a).

          4. Let R be the rotation of a vector by 45 degrees (?/4 radians) about the origin. Find the equation of the line produced by rotating the line y = -2x using the transformation R. You may take for granted that a linear transformation transforms a line into another line.
5. In this problem, the transformations R, D, and T are all linear transformations from R<sup>2</sup> to R<sup>2</sup>. You do not need to verify their linearity.
(a) Let R be the transformation that reflects elements of R<sup>2</sup> across the line y = x. Let D be the transformation that scales the vertical direction by 2/3 and scales the horizontal direction by 3/2. Find the matrix representations of R and D.
(b) Let T be given by $T\begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} -3 \ 3/2 \end{pmatrix}$ and $T\begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} -4 \ 3 \end{pmatrix}$. Find the matrix representation of T using the structure theorem.
(c) The transformation T can be decomposed into basic transformations. Namely, it is the dilation D, followed by a shear S, and lastly the reflection R. Find the matrix for the shear S. The transformations R and D are those from part (a).

4. Let R be the rotation of a vector by 45 degrees (?/4 radians) about the origin. Find the equation of the line produced by rotating the line y = -2x using the transformation R. You may take for granted that a linear transformation transforms a line into another line.
5. In this problem, the transformations R, D, and T are all linear transformations from R<sup>2</sup> to R<sup>2</sup>. You do not need to verify their linearity.
(a) Let R be the transformation that reflects elements of R<sup>2</sup> across the line y = x. Let D be the transformation that scales the vertical direction by 2/3 and scales the horizontal direction by 3/2. Find the matrix representations of R and D.
(b) Let T be given by T
< p m a t r i x >
=
< p m a t r i x > and T
< p m a t r i x >
=
< p m a t r i x >. Find the matrix representation of T using the structure theorem.
(c) The transformation T can be decomposed into basic transformations. Namely, it is the dilation D, followed by a shear S, and lastly the reflection R. Find the matrix for the shear S. The transformations R and D are those from part (a).

Added by Lawrence F.

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