Matrices as transformations Problem set A

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

1. (a) \(\begin{pmatrix}2 & -3 \\ 1 & 4\end{pmatrix}\), point \((4, 1)\)
2. (b) \(\begin{pmatrix}1 & 2 \\ -1 & 3\end{pmatrix}\), point \((-2, 5)\)
3. (c) \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\), point \((3, -7)\)

Write down the matrix for each transformation.

1. (a) Horizontal stretch, scale factor 5
2. (b) Vertical stretch, scale factor \(\frac{1}{3}\)
3. (c) Enlargement, scale factor −2
4. (d) Rotation 90° clockwise about the origin
5. (e) Rotation 30° anticlockwise about the origin
6. (f) Reflection in the line \(y = \sqrt{3}\,x\)

Find the point whose image is given.

1. (a) \(\begin{pmatrix}2 & -3 \\ 1 & 4\end{pmatrix}\), image \((1, 6)\)
2. (b) \(\begin{pmatrix}1 & 2 \\ -1 & 3\end{pmatrix}\), image \((8, 7)\)

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

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

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

Describe the transformation with each matrix.

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

(a) Find the matrix for a reflection in the line \(y = \frac{1}{3}x\). (b) Find the point whose image under this reflection is \((5, 5)\).