Standard vectors/Basis from linearly independent/Example

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

$v=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)\in K^n$

be an arbitrary vector. Then we have immediately

$v=\sum _{i=1}^n{b}_i{e}_i.$

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