Matrix Transformations Flashcards

All 16 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: $(3, -2)$

True or false? When combining two transformations, the two matrices can be multiplied in either order.

False. Matrix multiplication is not commutative, so $AB$ and $BA$ usually give different transformations.

What are the steps to describe geometrically the single transformation represented by $N^2$?

e.g. $N = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

1. Multiply the matrix by itself: $N^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$
2. Recognise the resulting matrix: rotation 180° about the origin (or enlargement scale factor $-1$ about the origin).

What are the steps to find the original point when you are given its image?

e.g. $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ maps $(x, y)$ to $(-2, 3)$

1. Set up the equation: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$
2. Multiply out for two equations: $-y = -2$ and $x = 3$
3. Solve: the point is $(3, 2)$

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 $y = -x$

What are the steps to find the matrix that returns a shape to its original position?
e.g. the shape has been enlarged, centre the origin, scale factor $2$

1. Describe the transformation that was applied: enlargement, centre origin, scale factor $2$
2. State the transformation that undoes it: enlargement, centre origin, scale factor $\frac{1}{2}$
3. Write down its matrix: $\begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix}$

True or false? Applying the same reflection matrix twice returns a shape to its original position.

True. A reflection is its own inverse, so the product of the matrix with itself is the identity.