Standard Basis of the Real Vector Space

TIP

The standard basis of the real vector space $({\mathbb{R}}^n,\mathbb{R},+,\cdot )$ is

$$\left(\underset{n\text{ vectors}}{\underbrace{{\vec{e}}_1=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right],{\vec{e}}_2=\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right],\cdots ,{\vec{e}}_n=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right]}}\right)$$

NOTATION

The standard basis ${\vec{e}}_1,{\vec{e}}_2,{\vec{e}}_3$ in ${\mathbb{R}}^3$ is often denoted as

$$\hat{ı}={\vec{e}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$$ $$\hat{ȷ}={\vec{e}}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$$ $$\hat{k}={\vec{e}}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

Theorem: Orthonormality of the Real Standard Basis

The standard basis of the real vector space $({\mathbb{R}}^n,\mathbb{R},+,\cdot )$ is orthonormal.

PROOF

TODO