Let $C _ { 1 } , \ldots , C _ { n }$ be $n$ column matrices in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$, with $C _ { 1 }$ non-zero. Prove that, if the family $\left( C _ { 1 } , \ldots , C _ { n } \right)$ is linearly dependent, then there exists a unique $j \in \llbracket 1 , n - 1 \rrbracket$ such that $$\left\{ \begin{array} { l }
\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent } \\
C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)
\end{array} \right.$$
Let $C _ { 1 } , \ldots , C _ { n }$ be $n$ column matrices in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$, with $C _ { 1 }$ non-zero.
Prove that, if the family $\left( C _ { 1 } , \ldots , C _ { n } \right)$ is linearly dependent, then there exists a unique $j \in \llbracket 1 , n - 1 \rrbracket$ such that
$$\left\{ \begin{array} { l }
\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent } \\
C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)
\end{array} \right.$$