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

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

• Observe that the matrix that you tried maps $(1,0)$ to $(1,1)$ and $(0,1)$ to $(0,0)$: it collapses the entire plane onto the line $x-y=0$.
– amd
Sep 7, 2019 at 19:21

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.

• sorry i dont quite understand, I have only started studying matrices for my GCSE
– yt.
Sep 7, 2019 at 17:19
• And what does GCSE mean? Sep 8, 2019 at 0:33

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 $$M\begin{pmatrix}0\\0\end{pmatrix},M\begin{pmatrix}1\\0\end{pmatrix},M\begin{pmatrix}0\\1\end{pmatrix},M\begin{pmatrix}1\\1\end{pmatrix}$$.