SOLUTION: Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u + v + w, v + w, w} is also a basis for V.

Algebra ->  College  -> Linear Algebra -> SOLUTION: Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u + v + w, v + w, w} is also a basis for V.       Log On


   

Question 3683: Let u, v, and w be distinct vectors of a vector space V. Show that if
{u, v, w} is a basis for V, then {u + v + w, v + w, w} is also a basis
for V.
Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
if a(u + v + w)+ b(v + w) + c w = 0 for scalars a,b,c
then a u + (a+b)v + (a+b+c)w = 0
since u,v,w are independent
we have a = 0, a+b = 0 and a+b+c = 0
This implies b = 0 and so c = 0.
This shows u + v + w, v + w, w are three independent
and so {u + v + w, v + w, w} forms a basis for V,
because {u,v,w} is a basis of V, dim V = 3.
Kenny