Basis exchange theorem/1/Example

We consider the standard basis ${\displaystyle {}e_{1},e_{2},e_{3}}$ of ${\displaystyle {}K^{3}}$ and the two linearly independent vectors ${\displaystyle {}u_{1}={\begin{pmatrix}3\\2\\1\end{pmatrix}}}$ and ${\displaystyle {}u_{2}={\begin{pmatrix}5\\4\\2\end{pmatrix}}}$. 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

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

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

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

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

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

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

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