\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}
$$ ## Finding the matrix for a particular (linear) transformation Observe that for a given matrix $$ A= \begin{bmatrix}a & b \\ c & d \end{bmatrix} $$The following is true:
$$a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a\\ c \end{bmatrix},$$ $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} b \\ d \end{bmatrix}$$ i.e. the $\hat{i}$ component will be $$\begin{bmatrix} a \\ c \end{bmatrix}$$ And the $\hat{j}$ component will be $$\begin{bmatrix} c \\ d \end{bmatrix}$$ Hence, **for a given linear transformation, you know the images of the points $(1,0)$ and $(0,1)$, you can easily find out all the possible translations!** [^1] [^2] ## Translations Recall that translations are **not** a linear transformation, as they translate the origin, which is not allowed for linear translations. Hoever, we can use matrixes to explain translation, by adding elements to the matrix: $$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix}$$ [^1]: Only *some* of these are on the data sheet. You will have to memorise the rest. [^2]: Shears are not in the syllabus. ur not in the syllabus $$
