(Solved): Prove If \( S \) is a basis for a \( v_{S} v \), then for any vectors \( U \) and \( v \) in \( v \ ...

Prove If \( S \) is a basis for a \( v_{S} v \), then for any vectors \( U \) and \( v \) in \( v \) and any Scalar \( K \), the following relatiouships hold. \[ \begin{array}{l} (u+v) s=u(s)+v(s) \\ (k v) s=k(v) s \end{array} \] Show if the following set of vectors form basis. \[ \begin{array}{c} v_{1}=(6,-3,5) \quad v_{2}=(2,7,5) \\ v_{3}=(14,6,6) \end{array} \] Assume that \( V \) consists of all vectors defined in \( R^{3}\left[\begin{array}{c}q+t \\ 2 r-t \\ 3 s+t\end{array}\right] * q, r_{1} s, t \) real *s