# Linear Algebra - Coordinate system

coordinate system in terms of vector.

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

$$v = \alpha_1 a_1 + \dots + \alpha_n a_n$$

• A vector v is represented by its coordinate representation $[\alpha_1, \dots , \alpha_n]$ . A vector v is represented by the vector $[\alpha_1, \dots , \alpha_n]$ of coefficients called the coordinate representation of v in terms of $a_1, \dots , a_n$ .

## 3 - Coordinate representation

The coordinate representation of v in terms of $a_1, \dots , a_n$ is the vector $[\alpha_1, \dots , \alpha_n]$ such that:

$$v = \alpha_1 a_1 + \dots + \alpha_n a_n$$

In this context, the coefficients are called the coordinates.

Example:

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

## 4 - Coordinate

### 4.1 - Definition

The coordinates are the coefficients $[\alpha_1, \dots , \alpha_n]$ 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 $$A = [ a_2, \dots a_n ]$$

• ${\bf u}$ The coordinate representation of ${\bf v}$ in terms of ${\bf a}_1, \dots , {\bf a}_n$ can be written as matrix-vector equation $A{\bf u} = {\bf v}$
• To go from a coordinate representation ${\bf u}$ to the vector being represented, we multiply A times ${\bf u}$ .
• To go from a vector ${\bf v}$ to its coordinate representation, we can solve the matrix-vector equation $A{\bf u} = {\bf v}$ .

### 4.2 - homogeneous

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

## 5 - How

### 5.1 - to ensure for each point only one coordinate representation

• How can we ensure that each point has only one coordinate representation?
• Answer: The generators $a_1, \dots ,a_n$ should form a basis.

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