# Linear Dependence of Equation Vectors

1. Oct 11, 2009

### pillanoid

1. The problem statement, all variables and given/known data

Determine whether the members of the given set of vectors are linearly independent for -$$\infty$$ < t < $$\infty$$. If they are linearly dependent, find the linear relation among them.

x(1)(t) = (e-t, 2e-t), x(2)(t) = (e-t, e-t), x(3)(t) = (3e-t, 0)

(the vectors are written as row vectors)

2. Relevant equations

The section in my book about linear dependence of equation vectors is immediately followed by a discussion of eigenvalues. Wronskians are not covered until the next chapter.

3. The attempt at a solution

I set up a matrix of equations, each vector in the problem statement a column of the matrix, augmented it with the 0 vector, and row-reduced, resulting in

[e-t, 0, -3e-t | 0; 0, e-t, 6e-t | 0]

The book doesn't give any examples, and I'm having a hard time with where to go from here, or if this is the right approach in the first place.

2. Oct 11, 2009

### Staff: Mentor

Has your book discussed linear independence and dependence for ordinary vectors? The idea here is about the same.

3. Oct 11, 2009

### LCKurtz

The e-t is just a red herring. You can factor it out and your problem amounts to asking whether (1,2), (1,1), and (3,0) are independent. You probably already know that you can't have 3 independent vectors in R2. So they are going to be dependent. If you try the condition directly for independence you check:

a(1,2) + b(1,1) + c(3,0) = 0 or

a + b + 3c = 0
2a + b + 0c = 0

The row reduction you have done is the same as writing these as

a + 0b -3c = 0
0a +b + 6c = 0

Clearly you can take the c on the other side and let it be most anything except 0. Say c = 1, giving a = 3 and b = -6.

So your row reduction really does the work, you just need to interpret it right.