Show that {u+v,au} and {au,bv} spanVand are sets of linearly independent vectors.

For example, for {u+v,au} …

Span: Letw=iu+jvbe a vector inV. Are there scalarsk,lsuch thatw=k(u+v)+l(au)? Well,k(u+v)+l(au) = (k+la)u+kv. It turns out that we can takek=jandk+la=i, i.e.l= (i−j)/a(sincea≠ 0). Hencew=j(u+v)+[(i−j)/a](au) is a linear combination ofu+vandau.

Linear independence: Supposei(u+v)+j(au) =0. Then (i+ja)u+iv=0. By linear independence ofuandv, we havei= 0 andi+ja= 0 ⇒ (asa≠ 0)j= 0. This shows thatu+vandauare linearly independent.

Repeat the same procedure for {au,bv}.