SOLUTION: Let W be a subset of R3 be the subspace spanned by the vectors w1 and w2 below. Find the coordinates of w with respect to the basis {w1, w2} (In particular, your answer will show

Algebra ->  Proofs -> SOLUTION: Let W be a subset of R3 be the subspace spanned by the vectors w1 and w2 below. Find the coordinates of w with respect to the basis {w1, w2} (In particular, your answer will show       Log On


   

Question 839172: Let W be a subset of R3 be the subspace spanned by the vectors w1 and w2 below. Find the coordinates of w with respect to the basis {w1, w2} (In particular, your answer will show that w is an element of W)
w1=(2, -1, 4)
w2=(5,2,-3)
w=(-6, -15, 40)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
w=a%2Aw%5B1%5D%2Bb%2Aw%5B2%5D
w%5Bx%5D=a%2A2%2Bb%2A5=-6
w%5By%5D=a%2A%28-1%29%2Bb%2A2=-15
1.2a%2B5b=-6
2.-a%2B2b=-15
Multiply eq.2 by 2 and add to eq. 1,
2a%2B5b-2a%2B4b=-6-30
9b=-36
b=-4
and
-a%2B2%28-4%29=-15
-a-8=-15
a=7
Now check to make sure that w%5Bz%5D is consistent with this solution.
w%5Bz%5D=a%2A4%2Bb%2A%28-3%29=40
%287%294%2B%28-4%29%28-3%29=40
28%2B12=40
40=40
True.
So then,
w=7w%5B1%5D-4w%5B2%5D
So w is a linear combination of the w%5B1%5D and w%5B2%5D and is therefore a member of W.