Transformations with Matrices Flashcards

All 11 cards in this deck

What is the $2\times2$ matrix for a rotation of 90° anticlockwise about the origin?

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

What is the $2\times2$ matrix for a rotation of 90° clockwise about the origin?

$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

This is the same as a rotation of 270° anticlockwise about the origin.

What is the $2\times2$ matrix for a rotation of 180° about the origin?

$\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

What is the $2\times2$ matrix for a reflection in the line $x = 0$?

$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

The line $x = 0$ is the $y$-axis.

What is the $2\times2$ matrix for a reflection in the line $y = 0$?

$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

The line $y = 0$ is the $x$-axis.

What is the $2\times2$ matrix for a reflection in the line $y = x$?

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

What is the $2\times2$ matrix for a reflection in the line $y = -x$?

$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

What is the $2\times2$ matrix for an enlargement, centre the origin, with scale factor $k$?

e.g. scale factor $3$

$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$

e.g. $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$

What do the two columns of a $2\times2$ transformation matrix represent?

The images of the unit square's points $(1, 0)$ and $(0, 1)$: column 1 is the image of $(1, 0)$, column 2 is the image of $(0, 1)$.

What are the steps to find the image of a point under a transformation matrix?

e.g. image of $(3, 2)$ under $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

1. Write the point as a column vector: $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
2. Multiply with the matrix on the left: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
3. Read off the image: $(3, -2)$

What are the steps to describe geometrically the single transformation represented by a $2\times2$ matrix?
e.g. $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

1. Read column 1 as the image of $(1, 0)$: $(0, -1)$
2. Read column 2 as the image of $(0, 1)$: $(-1, 0)$
3. Name the transformation those images give: reflection in the line $y = -x$