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

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

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 ${}K^{n}$.