Affine function and points I have an affine function $F:\mathbb{R} ^{2}\rightarrow \mathbb{R} $, such as $F\begin{bmatrix}
1\\ 
2
\end{bmatrix}$=3 .
Now, $\begin{bmatrix}
1\\ 
2
\end{bmatrix}$ is represented as a point; can i represent it in the Cartesian coordinate system ? If yes, this means that it behaves as a vector, however i know that points are different from vectors.
Where am i wrong ?
 A: Many operations you do on vectors aren't valid operations on points, because they depend on the choice of origin. That doesn't mean that anything will stop you from doing it. If you want to take a point $P = (x,y)$ and multiply it by $\frac12$ to get $(\frac x2, \frac y2)$, you can do that. It's just that you've implicitly found the midpoint between $P$ and the origin $(0,0)$. Someone else who places the origin at a different point (and has different coordinates for $P$ as a result) would get a different answer when they multiply $P$ by $\frac12$: they'd find the midpoint between $P$ and their origin.
If you want mathematics to enforce affine operations on you, then you should represent a point $(x,y)$ not as two-dimensional vector, but as the three-dimensional vector $$\begin{bmatrix} x \\ y \\ 1\end{bmatrix}.$$ An operation that takes a point and gives you a 3D vector with anything other than $1$ in the last coordinate is an operation that doesn't give you back a point. For example, if you have a matrix $$F = \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1\end{bmatrix}$$ then multiplying by $F$ takes points to points; if you have a matrix not of this form; it does not. So matrices whose last row is $\begin{bmatrix}0 & 0 & 1\end{bmatrix}$ are precisely the affine transformations.
(And if you take the difference of two points, you get a 2D vector: a 3D vector whose last coordinate is $0$. You can add vectors to points and get other points.)
