Write the matrix for the vertices of the above graphic and perform the following operation by using the corresponding matrix of the linear transformation.
- A rotation by 45 degrees followed by a translation which keeps the vertex (1, 1) fixed.
I know the rotation matrix and the translation matrix, but how to find the matrix for the vertices? Won't it be a 2x4 matrix? I won't be able to multiply it with the rotation/translation matrix.
Moreover, is the problem asking me to translate first or rotate first, because it uses the words 'followed by?
When matrices are used to encode linear transformations, points in $\Bbb R^2$ are encoded with $2 \times 1$ matrices. In this context, there is no such thing as a "translation matrix".
On the other hand, when matrices are used to encode affine transformations, points in $\Bbb R^2$ are encoded with $3 \times 1$ matrices. In particular, the point $(x,y)$ becomes the $3 \times 1$ matrix $(x,y,1)^T$. In your context, if $M$ encodes our affine transformation, then the product $M(x,y,1)^T$ yields the coordinates of the transformed points.
In this context, the vertices are $(1,1),(1,0),(2,2),(0,1)$. So, given a $3 \times 3$ matrix $M$, we can compute the transformed points with the product $$ M \ \pmatrix{1&1&2&0\\1&0&2&1\\1&1&1&1}. $$
To your second question, the problem is asking you to rotate first. The rotation occurs, and is then "followed by" a translation.