Can any $2\times 2$ matrix transform the unit square?

Question

Hi I am starting to learn about matrix transformations.

I am confused with vectors and the $2\times 2$ transformation matrices: - vectors can represent translation but $2\times 2$ transformations can't? I think they can only represent the other 3 transformations: reflection, rotation and enlargement right? - also with transformation matrices on a point $P$, I can turn $(x,y)$ into a $2\times 1$ vector and then multiply that by the transformation matrix to get the coordinates of P' ? But how can I manipulate the vector to translate a point $P$, I cant multiply the two $2\times 1$ vectors so do I add them together?

I am really confused between the two these $2\times 1$ ad $2\times 2$ matrices and how they can transform the unit square.

Also I'm not even sure if all $2\times 2$ matrix can transform the unit square, or can they only transform a point? $$\begin{pmatrix}1&0\\1&0\\\end{pmatrix}$$

for example I tried this matrix above with the unit square and I multiplied each of the points with the matrix

first I got A'= (1,1) C'= (0,0)

So I thought the matrix could be a rotation 90 degrees anticlockwise, center at the center of the unit square But then I transformed points $B$ and $O$ which did not support this $B'=(1,1)$ $O'=(0,0)$

Am I doing something wrong here because I was expecting $B'= (0,1)$ and $O'= (1,0)$ or do not all matrices work?

Thanks, sorry for asking too many questions

Answer 1

You seem to think that there are four types of linear transformations of $\mathbb R^2$: translations, reflections, rotation and enlargements. This is doubly wrong:

• translations are not linear;
• not all linear transformations are of one of those types.

And precisely the linear transformation associated with that matrix does not belong to any of those types.

Answer 2

Here are answers to some of your questions:

Every $2\times2$ matrix represents a linear transformation; that is, a map $f:\mathbb{R}^2\to\mathbb{R}^2$ with $f(x,y)=(ax+by,cx+dy)$ where $a,b,c,d$ are constants. Since a translation is not a linear map, it cannot be represented by a $2\times 2$ matrix.

To transform the unit square, start by looking at where each of the four vertices are going under the transformation. If the matrix representing the transformation is $M$, work out $M\begin{pmatrix}0\\0\end{pmatrix},M\begin{pmatrix}1\\0\end{pmatrix},M\begin{pmatrix}0\\1\end{pmatrix},M\begin{pmatrix}1\\1\end{pmatrix}$.