📂Linear Algebra

Order Basis and Coordinate Vectors

Definition¹

Let’s say $V$ is a finite-dimensional vector space. When a specific order is assigned to a basis of $V$, it is called an ordered basis.

Let’s say $\beta = \left\{ \mathbf{v}_{1}, \dots, \mathbf{v}_{n} \right\}$ is an ordered basis of $V$. Then, due to the uniqueness of basis representation, for $\mathbf{v} \in V$, scalars $a_{i}$ uniquely exist as follows.

$$\mathbf{v} = a_{1}\mathbf{v}_{1} + \dots a_{n}\mathbf{v}_{n}$$

$a_{1},\dots,a_{n}$ is called the coordinate of $\mathbf{v}$ relative to basis $\beta$. The matrix that has the $i$th coordinate as its $i$th component is called the coordinate vector of $\mathbf{v}$ relative to $\beta$ or coordinate matrix, and is denoted as $[\mathbf{v}]_{\beta}$.

$$[\mathbf{v}]_{\beta} = \begin{bmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{bmatrix}$$

Furthermore, the ordered basis $\beta$ is called a coordinate system.

Explanation

The basis is defined as a set, and the order in which the elements of the set are listed does not matter, which means $\alpha = \left\{ \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} \right\} = \left\{ \mathbf{e}_{2}, \mathbf{e}_{3}, \mathbf{e}_{1} \right\} = \beta$. Therefore, to abstract the concept of ‘coordinate’, it is necessary to assign order to the elements of the basis. Now, if we say $\alpha, \beta$ is an ordered basis,

$$\alpha = \left\{ \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} \right\} \ne \left\{ \mathbf{e}_{2}, \mathbf{e}_{3}, \mathbf{e}_{1} \right\} = \beta$$

• $[\mathbf{v}_{i}]_{\beta} = \mathbf{e}_{i}$ holds. $\mathbf{e}_{i}$ is the standard basis.

• The function $T : \mathbf{v} \mapsto [\mathbf{v}]_{\beta}$ becomes a linear transformation from $V$ to $\mathbb{R}^{n}$.

• Regarding vector space $\mathbb{R}^{n}$, $\left\{ \mathbf{e}_{1}, \dots, \mathbf{e}_{n} \right\}$ is called the standard ordered basis.