coordinate system in terms of vector.
Idea of coordinate system for a vector space V: (and generalized beyond two dimensions),
<MATH>v = \alpha_1 a_1 + \dots + \alpha_n a_n</MATH>
The coordinate representation of v in terms of <math>a_1, \dots , a_n</math> is the vector <math>[\alpha_1, \dots , \alpha_n]</math> such that:
<MATH>v = \alpha_1 a_1 + \dots + \alpha_n a_n</MATH>
In this context, the coefficients are called the coordinates.
Example:
The coordinates are the coefficients <math>[\alpha_1, \dots , \alpha_n]</math> of the Coordinate representation
Why put the coordinates in a vector? It makes sense to put the coordinates in a vector in view of linear-combinations definitions of matrix-vector multiplication. Let <MATH>A = [ a_2, \dots a_n ]</MATH>
Homogeneous coordinates are the following point representation: {'x','y','u'}-vector<math>\begin{bmatrix} x \\ y \\ u\end{bmatrix}</math> .