2
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ 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 $x-y=0$. $\endgroup$
    – amd
    Sep 7, 2019 at 19:21

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ sorry i dont quite understand, I have only started studying matrices for my GCSE $\endgroup$
    – yt.
    Sep 7, 2019 at 17:19
  • $\begingroup$ And what does GCSE mean? $\endgroup$ Sep 8, 2019 at 0:33
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.