Standard vectors/Basis from linearly independent/Example

The standard vectors in ${\displaystyle {}K^{n}}$ form a basis. The linear independence was shown in example. To show that they also form a generating system, let

${\displaystyle {}v={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}\in K^{n}\,}$

be an arbitrary vector. Then we have immediately

${\displaystyle {}v=\sum _{i=1}^{n}b_{i}e_{i}\,.}$

Hence, we have a basis, which is called the standard basis of ${\displaystyle {}K^{n}}$.