Standard Basis

The standard basis of ${\mathbb{R}}^n$ is made up of the vectors

$${\mathbf{e}}_1=\left[\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\end{array}\right],{\mathbf{e}}_2=\left[\begin{array}{c}0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right],\dots ,{\mathbf{e}}_n=\left[\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 1\end{array}\right]$$

Note: a standard basis only applies to vector spaces of ${\mathbb{R}}^n$

The set $\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_n\}$ forms a basis for ${\mathbb{R}}^n$ called the standard basis.

Each vector in ${\mathbb{R}}^n$ can be expressed as a linear combination of the standard basis vectors:

$$\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]=v_1{\mathbf{e}}_1+v_2{\mathbf{e}}_2+\cdots +v_n{\mathbf{e}}_n$$

We then say that $(v_1,v_2,\dots ,v_n)$ are the coordinate of $\vec{x}$ in the given matrix using the standard basis.