Basis exchange theorem/1/Example

We consider the standard basis ${}e_{1},e_{2},e_{3}$ of ${}K^{3}$ and the two linearly independent vectors ${}u_{1}={\begin{pmatrix}3\\2\\1\end{pmatrix}}$ and ${}u_{2}={\begin{pmatrix}5\\4\\2\end{pmatrix}}$. We want to extend this family to a basis, using the standard basis and according to the inductive method described in the proof of the basis exchange theorem. We first consider

${}u_{1}=3e_{1}+2e_{2}+e_{3}\,.$

Since no coefficient is ${}0$, we can extend ${}u_{1}$ with any two standard vectors to obtain a basis. We work with the new basis

$u_{1},e_{1},e_{2}.$

In a second step, we would like to include ${}u_{2}$. We have

${}u_{2}={\begin{pmatrix}5\\4\\2\end{pmatrix}}=2{\begin{pmatrix}3\\2\\1\end{pmatrix}}-e_{1}=2u_{1}-e_{1}+0e_{2}\,.$

According to the proof, we have to get rid of ${}e_{1}$, as its coefficient is ${}\neq 0$ in this equation (we can not get rid of ${}e_{2}$). The new basis is, therefore,

$u_{1},u_{2},e_{2}.$