Problem with linear dependence: det(a)=0, but rref is inconsistent?

[skyturnred]

Homework Statement

v1=[2 1 1 4 2]
v2=[-1 2 2 1 -1]
v3=[3 -2 1 -2 2]
v4=[4 1 4 3 3]
v5=[1 2 3 2 1]

Find if the system is linear dependent or independent. If it is dependent, express the last vector in the list (v5) as a combination of the preceding ones.

Homework Equations

The Attempt at a Solution

I am having trouble with this one. Using the definition of independence (c1V1+c2V2+...+cnVn=0), I get the following matrix:

a=[2 -1 3 4 1; 1 2 -2 1 2; 1 2 1 4 3; 4 1 -2 3 2; 2 -1 2 3 1]. When I find det(a), I find it to be 0. This means that the system has a non-trivial solution, right? And if so, that means that the system is linear dependent right? so then I take the same matrix as an augmented matrix with the final column ([1; 2; 3; 2; 1]) as being the augmented part, since it is equal to V5 and I am trying to express it as a combination of the others. Then I find rref. But when I find rref I get the following:

a(rref)=[2 0 0 1 0; 0 1 0 1 0; 0 0 1 1 0; 0 0 0 0 1], but looking at the very last row, the matrix is inconsistent. If it is inconsistent, there is no value for c1, c2, c3 and c4 that would equal V5. So where did I go wrong? I double checked all my steps so there must be a problem with my understanding.

 

[Office_Shredder]

It might not be that v5 is a linear combination of the others. For example, consider the vectors

v1=(1,0,0)
v2=(1,0,0)
v3=(0,1,0)

in R³. You can make a matrix and take its determinant to verify these are linearly dependent, but if you try to write v3 as a linear combination of v1 and v2 you're going to have trouble

 

[skyturnred]

It might not be that v5 is a linear combination of the others. For example, consider the vectors

v1=(1,0,0)
v2=(1,0,0)
v3=(0,1,0)

in R³. You can make a matrix and take its determinant to verify these are linearly dependent, but if you try to write v3 as a linear combination of v1 and v2 you're going to have trouble

OK, thanks. I thought that was the case. But the wording of the question I've been assigned is really horrible. It confused me.

Thanks again!

 

[Ray Vickson]

OK, thanks. I thought that was the case. But the wording of the question I've been assigned is really horrible. It confused me.

Thanks again!

If your wording of the problem was an exact copy of the wording you were given, it could not possibly have been clearer. Where did you see a problem?

RGV