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

vvv

Explanation:

Answer 2

`\{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`.