2.8a Basis

Basis

A set of vectors $B=\{{\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_n\}\in \mathcal{V}$ is a basis of the vector space $\mathcal{V}$ if:

1. The vectors ${\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_n$ are linearly independent.
2. The span of $B$ equals $\mathcal{V}$ (i.e., any vector $v\in \mathcal{V}$ can be built as a linear combination of $B$).

Intuition

A basis is not unique.

Example

Basis of Vector Space ${\mathbb{R}}^3$

Any vector in the vector space ${\mathbb{R}}^3$ can be built using the equation

$$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=a\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]+b\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]+c\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=a{\vec{e}}_1+b{\vec{e}}_2+c{\vec{e}}_3,$$

where ${\vec{e}}_n$ is the standard unit vector at position $n$.

Therefore, a basis of ${\mathbb{R}}^3$ is the set of vectors $\left\{\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\right\}.$