Matrix of the transformation (x,y)\to(ax+by,\,cx+dy)?

\begin{pmatrix}a&b\\ c&d\end{pmatrix}

Matrix of T with (1,0)\to(2,3) and (0,1)\to(-1,4)?

\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}

How do you find the image of (x,y) under a matrix \mathbf M?

Multiply: \mathbf M\begin{pmatrix}x\\ y\end{pmatrix}, with \mathbf M on the left.

Is \mathbf{AB}=\mathbf{BA} in general?

No, so the order of transformations matters.

How do you find \mathbf M if you know the images of two points that are not (1,0) and (0,1)?

Write \mathbf M\begin{pmatrix}x\\ y\end{pmatrix}= image for each point and solve the simultaneous equations.

Is (x,y)\to(x+2,y) a linear transformation?

No: it does not send the origin to the origin.

Linear transformations as matricesEdexcel International A Level Further Maths: Flashcards

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What does a linear transformation do to $(x,y)$?

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What does a linear transformation do to $(x,y)$ ?: Sends it to $(ax+by,\,cx+dy)$ .
Where does a linear transformation send the origin?: To the origin.
Matrix of the transformation $(x,y)\to(ax+by,\,cx+dy)$ ?: $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$
What is the first column of a transformation's matrix?: The image of $(1,0)$ .
What is the second column?: The image of $(0,1)$ .
Matrix of $T$ with $(1,0)\to(2,3)$ and $(0,1)\to(-1,4)$ ?: $\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}$
How do you find the image of $(x,y)$ under a matrix $\mathbf M$ ?: Multiply: $\mathbf M\begin{pmatrix}x\\ y\end{pmatrix}$ , with $\mathbf M$ on the left.
Image of $(3,2)$ under $\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}$ ?: $(4,17)$
Which matrix represents $B$ followed by $A$ ?: $\mathbf{AB}$
In $\mathbf{AB}$ , which transformation is applied first?: The one on the right, $B$ .
Is $\mathbf{AB}=\mathbf{BA}$ in general?: No, so the order of transformations matters.
How do you find $\mathbf M$ if you know the images of two points that are not $(1,0)$ and $(0,1)$ ?: Write $\mathbf M\begin{pmatrix}x\\ y\end{pmatrix}=$ image for each point and solve the simultaneous equations.
Is $(x,y)\to(x+2,y)$ a linear transformation?: No: it does not send the origin to the origin.

Exam questions on Linear transformations as matrices

A linear transformation $T$ of the plane maps $(1,0)$ to $(2,3)$ and $(0,1)$ to $(-1,4)$ .
Find the image of the point $(-2,5)$ under $T$ .2 marks
The matrix $\mathbf{R}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ represents the transformation $R$ and the matrix $\mathbf{S}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ represents the transformation $S$ .
Find the image of the point $(4,1)$ under $S$ followed by $R$ .2 marks
The linear transformation $T$ is represented by $\mathbf{T}=\begin{pmatrix}2&1\\ -1&3\end{pmatrix}$ and the linear transformation $U$ is represented by $\mathbf{U}=\begin{pmatrix}1&2\\ 0&-1\end{pmatrix}$ .
Find the matrix that represents $T$ followed by $U$ , and use it to find the image of the point $(1,2)$ under this combined transformation.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).