confused by notation in my computer vision book My book, Szeliski Computer Vision Algorithms and Applications says that:
$$
x' = [R t] \, \bar{x}
$$
and says that
$$
x' = Rx + t
$$
http://imgur.com/NVuQLGg
Can someone explain what $[R t]$ means, and what $\bar{x}$ means?
It says that $R$ is the $2\times 2$ rotation matrix, and $t$ is the translation.
I can't put my mind around how
$$
Rx + t = [R t] \, \bar{x}
$$
is $[R t]$ a legal notation in linear algebra/multidimensional calculus?
Can I make a matrix 
$$
U = 
\left[
\begin{matrix}
1 & 2 \\ 
3 & 4
\end{matrix}
\right]
$$
and a vector $v = (1, 2)$ and make a construct $G = [U v]$? 
I have never seen this kind of construct in my life. 
I understand it's a notation concern and it may be difficult to help me,
but if it's at all possible to explain what [Matrix vector] notation means if it's legal, i'd really appreciate your help.
 A: Looks to me like a notation to keep the translation part (affine map!) in an extra column.
$\bar{x}$ seems to be the homogenous version of vector $x$:
$$
\bar{x} = 
\left[
\begin{matrix} 
x_1 \\ 
x_2 \\
1
\end{matrix}
\right]
= 
\left[
\begin{matrix} 
x \\ 
1
\end{matrix}
\right]
$$
$2\times 3$ Notation:
$$
x' =
[ R t] \, \bar{x}
=
\left[
\begin{matrix} 
R_{11} & R_{12} & t_1 \\ 
R_{21} & R_{22} & t_2 \\
\end{matrix}
\right]
\left[
\begin{matrix} 
x_1 \\ 
x_2 \\
1
\end{matrix}
\right]
=
\left[
\begin{matrix} 
R_{11} x_1 + R_{12} x_2 + t_1 \\ 
R_{21} x_1 + R_{22} x_2 + t_2 \\
\end{matrix}
\right]
=
R x + t 
\in \mathbb{R}^2
$$
$3\times 3$ Notation:
$$
\bar{x}' =
\left[ 
\begin{matrix}
R & t \\
0^T & 1
\end{matrix}
\right] 
\, 
\bar{x}
=
\left[
\begin{matrix} 
R_{11} & R_{12} & t_1 \\ 
R_{21} & R_{22} & t_2 \\
0 & 0 & 1
\end{matrix}
\right]
\left[
\begin{matrix} 
x_1 \\ 
x_2 \\
1
\end{matrix}
\right]
=
\left[
\begin{matrix} 
R_{11} x_1 + R_{12} x_2 + t_1 \\ 
R_{21} x_1 + R_{22} x_2 + t_2 \\
1
\end{matrix}
\right]
=
\left[
\begin{matrix} 
R x + t \\ 
1
\end{matrix}
\right]
\in \mathbb{R}^3
$$
Going from $2\times 2$ to $2\times 3$ matrices (extra column) allows to represent the translation transformations as well.
The idea behind homogenous cooordinates is to add an extra dimension (extra column and extra row) which allows to represent all transformations as matrices and the composition of several transformations as matrix products.
You can chain two $3\times 3$ matrices by matrix multiplication, but not two
$2\times 3$ matrices.
See also the slight uglyness with $2\times 3$ notation that it is $[R t] \bar{x}$ and not $[R t] x$ because a 3D vector is required. 
In case of 2D vectors and $2\times 2$ matrices you end up with 3D vectors and $3\times 3$ matrices. The third component of the vectors is usually normalized to $1$. See this article for some examples.
OpenGL is an example of a 3D graphics library / hardware using $4\times 4$ matrices.
