Let u and v be two vectors in Rn Show that the subspace span

Let u and v be two vectors in R^n. Show that the subspace spanned by the set {u, v} is the same as the subspace spanned by {u + v, u - v}

Solution

Since u =1/2(u+v ) +1/2(u-v) and v = 1/2(u+v)-1/2(u-v), hence every linear combination of u and v can also be expressed as a linear combination of u+v and u-v.

The subspace spanned by u and v contains all linear combinations of u and v. Also, the subspace spanned by u+v and u -v contains all linear combinatuions of u+v and u-v. Since u and v are themselves linear combinations of u+v and u-v ( and apparently u+v and u-v are linear combinations of u and v), hence Tte subspace spanned by u and v is the same as the subspace spanned by u+v and u-v.