Question: Let P be a vector space generated by the set:

\[S=\{\left( 0,1,1 \right),\left( 1,2,3 \right),\left( 3,-1,2 \right)\}\]

and Q be the vector space generated by

\[T=\{\left( 1,1,0 \right),\left( 0,1,1 \right)\}\]

Is S a basis for P? If not, construct a basis.

Is T a basis for Q? If not, construct a basis.

Find the intersection of P and Q. Is the intersection a vector space?