Question

Let u, v, and w be distinct vectors of a vector space V. Prove that if {u, v, w} is a basis for V, then {u + v, u + w, v + w} is also a basis for V. (Note: V is an arbitrary vector space, so you may not assume that u, v, and w are tuples.)

Answer 1

$\{u,v,w\}$ forms a basis for $V$, which means any vector $x\in V$ can be written as the linear combination,

$x=c_1u+c_2v+c_3w$

Pick $c_1=d_1+d_2$, $c_2=d_1+d_3$, and $c_3=d_2+d_3$; then

$x=(d_1+d_2)u+(d_1+d_3)v+(d_2+d_3)w$

$x=d_1(u+v)+d_2(u+w)+d_3(v+w)$

Any $x\in V$ can thus be written as a linear combination of the vectors $\{u+v,u+w,v+w\}$, so these three vectors also form a basis for $V$.