Answer
a) $(5 t,-3 t, t, t) =t(5,-3,1,1)$, hence $S=\{(5,-3,1,1)\}$ is a basis for $W$.
b) The dimension of $W$ is $1$.
Work Step by Step
Assume the subspace $W=\{(5 t,-3 t, t, t) : t \text { is a real number }\}$, then
a) $(5 t,-3 t, t, t) =t(5,-3,1,1)$, hence $S=\{(5,-3,1,1)\}$ is a basis for $W$.
b) The dimension of $W$ is $1$.
