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.Com
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)   (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

RELATED QUESTIONS

show that if u+v=u+w then v=w , where u,v,w belongs to vector... (answered by ankor@dixie-net.com)
Let u = (2,1,2), v = (3,2,1) and w = (1,2,-5) be vectors in 3-dimension space. (a).... (answered by Alan3354)
(u+w)²-v²÷(v+w)²-u² (answered by robertb)
If u, v and w are unit vectors and satisfy condition u + v + w = 0 then find u.v + u.w +... (answered by ikleyn)
If (w^4 v^5)/(u^3) is greater than 0, which of the following does not have to be... (answered by greenestamps)
Each of angles b/t the vectors u,v,& w is 60. ||u||= 4. . ||v||=2 & ||w||=6 Then find... (answered by robertb)
1)u×u>=0 and u×u=0 if and only if u=0 prove theorem? 2)Show that there are no vectors u... (answered by lynnlo)
Please help me ㅠㅠ help me 1)u×u>=0 and u×u=0 if and only if u=0 prove... (answered by lynnlo)
1)u×u>=0 and u×u=0 if and only if u=0 prove theorem? 2)Show that there are no vectors u... (answered by lynnlo)