# Description

For a subspace $$V$$, we define a basis $$\mathcal{B}$$ as an **ordered** set of vectors $${\vec{v}\_1, \vec{v}\_2, \ldots, \vec{v}\_n}$$, such that

1. $$\text{span}{\vec{v}\_1, \vec{v}\_2, \ldots, \vec{v}\_n} = V$$ and
2. $${\vec{v}\_1, \vec{v}\_2, \ldots, \vec{v}\_n}$$ is linearly independent.

Because of these properties, we can express any vector in the vector space $$V$$ as a linear combination of the basis vectors.

> **Note**: The order of the basis vectors matters for a basis!

If we don't specify a basis, we are usually referring to the standard basis $$\mathcal{S} = \left{\begin{bmatrix}1 \ 0 \ 0 \ \vdots \ 0\end{bmatrix}, \begin{bmatrix}0 \ 1 \ 0 \ \vdots \ 0\end{bmatrix}, \ldots, \begin{bmatrix}0 \ 0 \ 0 \ \vdots \ 1\end{bmatrix}\right}$$.

In fact, we have been almost always using the standard basis vectors. For example, let $$\vec{v} = \begin{bmatrix}5 \ 4 \ 2\end{bmatrix}$$ in the standard basis.

$$\vec{v} = \begin{bmatrix}5 \ 4 \ 2\end{bmatrix} = 5\begin{bmatrix}1 \ 0 \ 0\end{bmatrix} + 4\begin{bmatrix}0 \ 1 \ 0\end{bmatrix} + 2\begin{bmatrix}0 \ 0 \ 1\end{bmatrix}$$

To be precise, we therefore write

$$\[\vec{v}]\_\mathcal{S} = \begin{bmatrix}5 \ 4 \ 2\end{bmatrix}$$