We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$. Prove that, for all $\omega \in \Omega$, the family $\left( C ( \omega ) , C ^ { \prime } ( \omega ) \right)$ is linearly dependent if and only if there exists $\varepsilon \in \{ - 1,1 \}$ such that $C ^ { \prime } ( \omega ) = \varepsilon C ( \omega )$.
We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$.
Prove that, for all $\omega \in \Omega$, the family $\left( C ( \omega ) , C ^ { \prime } ( \omega ) \right)$ is linearly dependent if and only if there exists $\varepsilon \in \{ - 1,1 \}$ such that $C ^ { \prime } ( \omega ) = \varepsilon C ( \omega )$.