Question

let v=R^3 be a vector space over the field R. if V has basis B={u,v,w}. then u,v,w are​

Answer 1

Answer:

Let u,v,w∈V a vector space over a field F such that u≠v≠w. Let {u,v,w} be a basis for V. Because {u,v,w} is a basis, then u,v,w are linearly independent and ⟨{u,v,w}⟩=V.

Let x∈V be an arbitrary vector then x can be uniquely expressed as a linear combination of {u,v,w}. Let's suppose x=au+bv+cw for some a,b,c∈F.

On the other hand, let us consider {u+v+w,v+w,w}⊆V.

Then

⟨{u+v+w,v+w,w}⟩={d(u+v+w)+e(v+w)+f(w)∣d,e,f∈F}={du+(d+e)v+(d+e+f)w∣d,e,f∈F}.

If x∈V, then x=du+(d+e)v+(d+e+f)w is another unique representation of x∈V . Then for any arbitrary x∈V, we have d=a, d+e=band d+e+f=c∈F.

Because {u,v,w} is a basis for V, then {u+v+w,v+w,w} must also be a basis for V.