# Rotations and translations transformations

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?

• What is a translation matrix? I ask because translation is not linear in the sense of linear algebra ...
– Levi
Nov 10, 2019 at 10:57
• @Levi Perhaps the asker is using homogeneous coordinates to apply affine transformations with matrices. Nov 10, 2019 at 11:00

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 $$M \ \pmatrix{1&1&2&0\\1&0&2&1\\1&1&1&1}.$$
• It looks like you're trying to ask "Won't $\pmatrix{1&1&2&0\\1&0&2&1}$ be the vertex matrix"? That depends on what exactly your book defines a "vertex matrix" to be. However, if we want to use a $3 \times 3$ matrix to apply the transformation, then we need the row of $1$s on the bottom. Nov 10, 2019 at 11:21