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

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