Let v and w be two vectors of R^n. Show that span{v,w}=Span{v+w,v-w}?

Question

(they all have vector arrows over them)

Answer 1

span{v,w}=all vectors of form av+bw

av+bw=((a+b)/2)(v+w)+((a-b)/2)(v-w)

c(v+w)+d(v-w)=(c+d)v+(c-d)w

Hence, any linear combinatin of one pair can be presented as a linear combination of the other pair, which proves span{v,w}=span{v+w,v-w}