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enter image description here

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?

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

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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.

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    $\begingroup$ 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. $\endgroup$ Nov 10, 2019 at 11:21

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