independent or dependent: {x^2-1, x^2+1, 4x, 2x - 3}

[calchere]

Determine whether the given set of vectors is dependent or independent.

{x^2-1, x^2+1, 4x, 2x-3}

So, it is dependent if:
r1v1 + r2v2 +...+rkvk=0 and some rj does not = 0

r(x^2-1) + s(x^2+1) + t(4x) + u(2x-3) = 0

I created two equations. One with x=0 and one with x=1.

x=0
r(-1) + s(1) + t(0) + u(-3)
x=1
r(0) + s(2) + t(4) + u(-1)

Now, if i use r=1, s=1, t= -1/2 u=0 for both equations, they both equal zero, so it would be dependent. However, if i plug these numbers into the orignal equation, it does not equal zero. The answer IS dependent, but i don't think i worked the problem correctly. Any help would be appreciated.

Btw, this is a linear algebra problem. i wasn't sure which section i was suppost to post it under.

 

[galactus]

Set up the problem:

\(\displaystyle \L\\a(x^{2}-1)+b(x^{2}+1)+c(4x)+d(2x-3)=0\)

Equate coefficients:

a + b = 0
4c+2d =0
-a+b-3d=0

We have 3 equations with 4 unknowns, which is the same as saying we have 4 vectors in R^3.

Therefore, it's dependent.