Homogeneous coordinates

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Rotation and scaling of n-dimensional coordinates can be described by n-by-n matrices. Conveniently consider two dimendions, so $n=2$. Each of the 4 components of the matrix is multiplied by one of the two coordinates of the vector:

$\left[\begin{array}{c} x_1\\ y_1 \end{array}\right]= \mathbf{T}\cdot \left[\begin{array}{c} x_0\\ y_0 \end{array}\right]= \left[\begin{array}{cc} t_{11} & t_{12}\\ t_{21} & t_{22} \end{array}\right] \cdot \left[\begin{array}{c} x_0\\ y_0 \end{array}\right]= \left[\begin{array}{cc} t_{11}\cdot x_0+t_{12}\cdot y_0\\ t_{21}\cdot x_0+t_{22}\cdot y_0 \end{array}\right]$

The same holds for higher dimensions. All the components of the matrix are multiplied with one of the vectors $n$ components. So the transformations are dependent on the original coordinates. Translation however is actually a vector addition and so independent of the original coordinates (see subarticle about translation). To be able to also apply a translation by matrix multiplication, an additional dimension is introduced. If the actual space is n-dimensional, the transformation matrix is an (n+1)-by-(n+1) matrix. The last row (with index n+1) always contains zeros and a 1 at the end. The additional component of vectors is always a 1. So the multiplication for $n=2$ for example results in:

$\left[\begin{array}{c} x_1\\ y_1\\ 1 \end{array}\right]= \mathbf{T} \cdot \left[\begin{array}{c} x_0\\ y_0\\ 1 \end{array}\right]= \left[\begin{array}{ccc} t_{11} & t_{12} & t_{13}\\ t_{21} & t_{22} & t_{23}\\ 0 & 0 & 1 \end{array}\right] \cdot \left[\begin{array}{c} x_0\\ y_0\\ 1 \end{array}\right]= \left[\begin{array}{c} t_{11}\cdot x_0+t_{12}\cdot y_0+t_{13}\cdot 1\\ t_{21}\cdot x_0+t_{22}\cdot y_0+t_{23}\cdot 1\\ 0\cdot x_0+0\cdot y_0+1\cdot1 \end{array}\right]= \left[\begin{array}{c} t_{11}\cdot x_0+t_{12}\cdot y_0+{\color{Green}\mathbf{t_{13}}}\\ t_{21}\cdot x_0+t_{22}\cdot y_0+{\color{Green}\mathbf{t_{23}}}\\ 1 \end{array}\right]$

As can be seen in the resulting vector, the components $t_{13}$ and $t_{23}$ are independent of the original coordinates and just added to the x- or the y-coordinate, respectively.