linear_algebra:coordinate

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Idea of coordinate system for a vector space V: (and generalized beyond two dimensions),

Coordinate system for a vector space V is specified by generators <math>a_1, \dots, a_n</math> of V
Every vector v in V can be written as a linear combination

<MATH>v = \alpha_1 a_1 + \dots + \alpha_n a_n</MATH>

A vector v is represented by its coordinate representation <math>[\alpha_1, \dots , \alpha_n]</math> . A vector v is represented by the vector <math>[\alpha_1, \dots , \alpha_n]</math> of coefficients called the coordinate representation of v in terms of <math>a_1, \dots , 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:

Coordinate representation of the vector <math>v = [1, 3, 5, 3]</math> in terms of the vectors <math>[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]</math> . Since the vector is equal to <math>1 [1, 1, 0, 0] + 2 [0, 1, 1, 0] + 3 [0, 0, 1, 1]</math> the coordinate representation of v is <math>[1, 2, 3]</math>
Coordinate representation of the vector <math>[6, 3, 2, 5]</math> in terms of the vectors <math>[2, 2, 2, 3], [1, 0,-1, 0], [0, 1, 0, 1]</math> : Since <math>[6, 3, 2, 5] = 2 [2, 2, 2, 3] + 2 [1, 0,-1,0]-1 [0, 1, 0, 1]</math> the coordinate representation is <math>[2, 2,-1]</math>
Coordinate representation of the vector <math>[0,0,0,1]</math> in terms of the vectors <math>[1,1,0,1], [0,1,0,1], [1,1,0,0]</math> over Gf2 Since <math>[0, 0, 0, 1] = 1 [1, 1, 0, 1] + 0 [0, 1, 0, 1] + 1 [1, 1, 0, 0]</math> the coordinate representation of <math>[0, 0, 0, 1]</math> is <math>[1, 0, 1]</math>

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>

<math>{\bf u}</math> The coordinate_representation of <math>{\bf v}</math> in terms of <math>{\bf a}_1, \dots , {\bf a}_n</math> can be written as matrix-vector equation <math>A{\bf u} = {\bf v}</math>

To go from a coordinate representation <math>{\bf u}</math> to the vector being represented, we multiply A times <math>{\bf u}</math> .
To go from a vector <math>{\bf v}</math> to its coordinate representation, we can solve the matrix-vector equation <math>A{\bf u} = {\bf v}</math> .

Homogeneous coordinates are the following point representation: {'x','y','u'}-vector<math>\begin{bmatrix} x \\ y \\ u\end{bmatrix}</math> .