Question 22330
 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