Write down the \( 2\times 2 \) matrix that represents a reflection in the line \( y = x \).
\( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
Linear Transformations
12 exam-style questions on 2D/3D transformations, invariant lines/points and successive maps. Tap Show answer for full working.
4 Easy 4 Medium 4 Hard
Find the matrix that represents an anticlockwise rotation through \( 90^\circ \) about the origin.
\( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
The matrix \( \mathbf{A} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \) represents a transformation. Describe the geometric effect of the transformation fully.
Stretch scale factor 2 parallel to the \( x \)-axis and stretch scale factor 3 parallel to the \( y \)-axis (or enlargement with different scale factors along the axes).
Find the image of the point \( (3, -1) \) under the transformation represented by \( \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \).
\( \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \). Image is \( (1, -1) \).
The transformation \( T \) is represented by \( \mathbf{M} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \). Describe \( T \) geometrically and find the image of the unit square with vertices \( (0,0) \), \( (1,0) \), \( (1,1) \), \( (0,1) \).
\( T \) is a clockwise rotation through \( 90^\circ \) about the origin (or anticlockwise \( 270^\circ \)).
Images: \( (0,0)\to(0,0) \), \( (1,0)\to(0,-1) \), \( (1,1)\to(1,-1) \), \( (0,1)\to(1,0) \).
Find the \( 2\times 2 \) matrix that represents a reflection in the line \( y = -x \), followed by an enlargement scale factor 2 centred at the origin.
Reflection in \( y=-x \): \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \). Enlargement SF 2: \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \).
Combined (reflection first): \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ -2 & 0 \end{pmatrix} \).
The matrix \( \mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix} \) represents a linear transformation. Find the invariant lines through the origin.
Invariant lines satisfy \( \mathbf{A}\begin{pmatrix}x\\y\end{pmatrix} = \lambda \begin{pmatrix}x\\y\end{pmatrix} \). Eigenvalues: \( \det(A-\lambda I)=(3-\lambda)^2-1=0 \) ⇒ \( \lambda=4 \) or \( \lambda=2 \).
For \( \lambda=4 \): \( (A-4I)\mathbf{v}=\begin{pmatrix}-1&1\\1&-1\end{pmatrix}\mathbf{v}=0 \) ⇒ \( y=x \). Line \( y=x \).
For \( \lambda=2 \): \( (A-2I)=\begin{pmatrix}1&1\\1&1\end{pmatrix} \) ⇒ \( y=-x \). Line \( y=-x \).
Show that the transformation represented by \( \mathbf{B} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \) maps the whole plane onto a single line, and find the equation of that line.
\( \det B = 4-4=0 \), singular. Image of \( (x,y) \) is \( (x+2y, 2x+4y) = (x+2y)(1,2) \). All image points lie on the line \( Y=2X \).
The transformation \( T \) is represented by \( \mathbf{M} = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} \). Find all invariant points and all invariant lines of \( T \).
Invariant points: \( \mathbf{M}\mathbf{r}=\mathbf{r} \) ⇒ \( (M-I)\mathbf{r}=0 \) ⇒ \( \begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=0 \) ⇒ \( y=0 \) and \( x+y=0 \) ⇒ only the origin.
Invariant lines through origin: eigenvalues of \( M \) are both 2 (repeated). Eigenspace: \( (M-2I)=\begin{pmatrix}0&1\\0&0\end{pmatrix} \) ⇒ \( y=0 \). Only the \( x \)-axis is an invariant line through the origin.
(Non-origin lines of the form \( y=c \) are mapped to \( y=2c \), so only \( c=0 \).)
Find the matrix that represents a rotation through \( \theta \) anticlockwise about the origin, followed by a reflection in the \( x \)-axis. Hence find the single matrix for \( \theta = \pi/3 \).
Rotation: \( R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \). Reflection in \( x \)-axis: \( F = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \).
Combined (rotation first): \( F R = \begin{pmatrix} \cos\theta & -\sin\theta \\ -\sin\theta & -\cos\theta \end{pmatrix} \).
For \( \theta=\pi/3 \): \( \begin{pmatrix} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{pmatrix} \).
The linear transformation \( T \) in the plane is given by \( \mathbf{r}' = \mathbf{A}\mathbf{r} \) where \( \mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix} \). Find the equation of the image of the line \( x + 2y = 1 \) under \( T \).
\( \det A=0 \), so image of the plane is the line through the origin in direction \( (1,3) \) (column space). Points on \( x+2y=1 \) map to \( \begin{pmatrix}x+2y\\3x+6y\end{pmatrix} = (x+2y)\begin{pmatrix}1\\3\end{pmatrix} = 1\cdot\begin{pmatrix}1\\3\end{pmatrix} \). Every point on the original line maps to the single point \( (1,3) \). The image is the single point \( (1,3) \) (degenerate).
In 3D, the matrix \( \mathbf{P} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} \) represents a transformation. Describe the geometric effect of \( P \) and find the image of the plane \( z=0 \).
\( P \) fixes the \( x \)-axis and rotates the \( yz \)-plane by \( 90^\circ \) ( \( (y,z)\mapsto(-z,y) \) ), i.e. a rotation of \( 90^\circ \) about the \( x \)-axis (from \( y \) toward \( z \) or check orientation).
Plane \( z=0 \): points \( (x,y,0) \) map to \( (x, 0, y) \). Image is the plane \( y=0 \) (the \( xz \)-plane).