If u1u2u3 span V some vector space Then do v1 u1v2 u2 u1v3

If u1,u2,u3 span V (some vector space). Then do v1 = u1,v2 = u2 u1,v3 = u3 span V? Show your work.

Let, x belong to span of u1,u2,u3

So there exist scalars,a,b,c so that

x=au1+bu2+cu3

x=au1+b(u2-u1+u1)+cu3

x=au1+b(u2-u1)+bu1+cu3

x=(a+b)u1+b(u2-u1)+cu3

Hence, x is also in span of v1,v2,v3

Let y be in span of v1,v2,v3

So there exist scalars r,s,t so that

y=rv1+sv2+tv3

y=ru1+s(u2-u1)+tu3

y=(r-s)u1+su2+tu3

Hence, y is in span of u1,u2,u3

Hence,

span{u1,u2,u3} =span{v1,v2,v3}