Matrices as transformations Problem set B

Find the image of each point under the transformation with the given matrix.

1. (a) \(\begin{pmatrix}3 & -1 \\ 2 & 5\end{pmatrix}\), point \((2, 3)\)
2. (b) \(\begin{pmatrix}2 & 1 \\ -3 & 4\end{pmatrix}\), point \((-3, 2)\)
3. (c) \(\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\), point \((-6, 2)\)

Write down the matrix for each transformation.

1. (a) Horizontal stretch, scale factor 4
2. (b) Vertical stretch, scale factor \(\frac{1}{2}\)
3. (c) Enlargement, scale factor −3
4. (d) Rotation 90° anticlockwise about the origin
5. (e) Rotation 45° clockwise about the origin
6. (f) Reflection in the line \(y = \frac{1}{\sqrt{3}}\,x\)

Find the point whose image is given.

1. (a) \(\begin{pmatrix}3 & -2 \\ 1 & 4\end{pmatrix}\), image \((4, 6)\)
2. (b) \(\begin{pmatrix}2 & 1 \\ -1 & 3\end{pmatrix}\), image \((6, 11)\)

The unit square has vertices \(O(0,0),\ A(1,0),\ C(1,1),\ B(0,1)\). Find the vertices of its image under each matrix, and sketch it.

1. (a) \(\begin{pmatrix}2 & 1 \\ 1 & 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}0 & 3 \\ -3 & 0\end{pmatrix}\)
3. (c) \(\begin{pmatrix}1 & -1 \\ 2 & 1\end{pmatrix}\)

Each transformation sends \((1,0)\) and \((0,1)\) to the points given. Write down its matrix.

1. (a) \((5, -1)\) and \((2, 3)\)
2. (b) \((0, -4)\) and \((4, 0)\)
3. (c) \((2, 0)\) and \((-3, 1)\)

Describe the transformation with each matrix.

1. (a) \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)
2. (b) \(\begin{pmatrix}-3 & 0 \\ 0 & -3\end{pmatrix}\)
3. (c) \(\begin{pmatrix}5 & 0 \\ 0 & 5\end{pmatrix}\)
4. (d) \(\begin{pmatrix}0.8 & -0.6 \\ 0.6 & 0.8\end{pmatrix}\)

(a) Find the matrix for a reflection in the line \(y = 2x\). (b) Find the point whose image under this reflection is \((10, 5)\).