SOLUTION: Let v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2). thank you

Algebra ->  College  -> Linear Algebra -> SOLUTION: Let v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2). thank you      Log On


   

Question 22330: Let v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2). thank you
Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2).
Try to solve
aw1 + bw2 = v1,
cw1 + dw2 = v2,
and
ew1 + fw2 = v3.
i.e. a(1,-2,-5)+b(0,8,9)=( 1,6,4) ...(1)
c(1,-2,-5)+d(0,8,9)=(2,4,-1)...(2)
e(1,-2,-5)+ f(0,8,9)=(-1,2,5)...(3)
(1) becomes:
a+0=1, -2a+8b = 6, -5a+9b = 4.
So,a=-1, b=1.
Solving (b),(c) left for you to see if they have feasible
solutions. so, then v1,v2,v3 are in the span {w1,w2}.

Next. since w1, w2 are indep, dim spn(w1,w2} =2.
Also, dim span(v1,v2,v3) >=2.
Hence, span(v1,v2,v3)=span(w1,w2)
Kenny