Matrix for a rotation of \theta anticlockwise about the origin?

\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}

Matrix for reflection in y=x?

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

Matrix for reflection in y=-x?

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

Matrix for an enlargement, scale factor k, centre the origin?

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

Matrix for a shear with the x-axis invariant, sending (0,1) to (\lambda,1)?

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

Matrix of A followed by B?

\mathbf{B}\mathbf{A} (the first transformation is on the right).

Is the order of successive transformations important?

Yes: matrix multiplication is not commutative, so \mathbf{B}\mathbf{A}\neq\mathbf{A}\mathbf{B} in general.

Matrix for reflection in the plane x=0?

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

Matrix for a rotation through \theta about the z-axis?

\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}

Matrix for a rotation through \theta about the y-axis?

\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}

Which direction is a positive rotation about a coordinate axis?

Anticlockwise when looking from the positive axis towards the origin (right-hand rule).

Linear transformationsAQA A-Level Further Maths: Flashcards

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Where do the columns of a transformation matrix come from?

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Where do the columns of a transformation matrix come from?: They are the images of the unit vectors $(1,0)$ and $(0,1)$ (and $(0,0,1)$ in 3D).
Matrix for a rotation of $\theta$ anticlockwise about the origin?: $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$
Matrix for reflection in $y=x$ ?: $\begin{pmatrix}0&1\\1&0\end{pmatrix}$
Matrix for reflection in $y=-x$ ?: $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$
Matrix for an enlargement, scale factor $k$ , centre the origin?: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$
Matrix for a shear with the $x$ -axis invariant, sending $(0,1)$ to $(\lambda,1)$ ?: $\begin{pmatrix}1&\lambda\\0&1\end{pmatrix}$
Matrix of $A$ followed by $B$ ?: $\mathbf{B}\mathbf{A}$ (the first transformation is on the right).
Is the order of successive transformations important?: Yes: matrix multiplication is not commutative, so $\mathbf{B}\mathbf{A}\neq\mathbf{A}\mathbf{B}$ in general.
Matrix for reflection in the plane $x=0$ ?: $\begin{pmatrix}-1&0&0\\0&1&0\\0&0&1\end{pmatrix}$
Matrix for a rotation through $\theta$ about the $z$ -axis?: $\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}$
Matrix for a rotation through $\theta$ about the $y$ -axis?: $\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}$
Which direction is a positive rotation about a coordinate axis?: Anticlockwise when looking from the positive axis towards the origin (right-hand rule).

Exam questions on Linear transformations

The matrix $\mathbf{M}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ represents a transformation $T$ of the plane.
The point $P$ is mapped by $T$ to $(-4,5)$ . Find the coordinates of $P$ .2 marks
The unit square $OABC$ has vertices $O(0,0)$ , $A(1,0)$ , $B(1,1)$ and $C(0,1)$ . It is transformed by the matrix $\mathbf{N}=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ .
Describe fully the single transformation represented by $\mathbf{N}^2$ .2 marks
Transformation $P$ is a rotation through $90^\circ$ clockwise about the origin, and transformation $Q$ is a reflection in the line $y=x$ .
Find the single matrix that represents $P$ followed by $Q$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).