Standard vectors/Linearly independent/Example

The standard vectors in ${\displaystyle {}K^{n}}$ are linearly independent. A representation

${\displaystyle {}\sum _{i=1}^{n}s_{i}e_{i}=0\,}$

just means

${\displaystyle {}s_{1}{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}+s_{2}{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}+\cdots +s_{n}{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}\,.}$

The ${\displaystyle {}i}$-th row yields directly ${\displaystyle {}s_{i}=0}$.