Expressing a non-trivial linear dependency relationship

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September 8th 2009, 04:53 AM
drew.walker

Expressing a non-trivial linear dependency relationship

I've been ask to see if a set of vectors is linearly independent, and if not, find a non-trivial linear dependency relationship.

I believe I've correctly determined that the set is linearly dependent and found the solution, but I don't know how I'm meant to express the linearly dependency relationship??

This is what I have so far:

$ v_1 = \begin{pmatrix}2 \\ 1 \\ 2\end{pmatrix} v_2 = \begin{pmatrix}4 \\ 4 \\ 5\end{pmatrix} v_3 = \begin{pmatrix}6 \\ 1 \\ 5\end{pmatrix} $

As a matrix, then reducing:

$ \begin{bmatrix} 2 & 4 & 6 \\ 1 & 4 & 1 \\ 2 & 5 & 5 \end{bmatrix} $

$ \begin{bmatrix} 1 & 0 & 5 \\ 1 & 4 & 1 \\ 2 & 5 & 5 \end{bmatrix} $

$ \begin{bmatrix} 1 & 0 & 5 \\ 1 & 4 & 1 \\ 0 & -3 & 3 \end{bmatrix} $

$ \begin{bmatrix} 1 & 0 & 5 \\ 1 & 1 & 4 \\ 0 & -3 & 3 \end{bmatrix} $

$ \begin{bmatrix} 1 & 0 & 5 \\ 0 & 1 & -1 \\ 0 & -3 & 3 \end{bmatrix} $

$ \begin{bmatrix} 1 & 0 & 5 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix} $

If we let $x_4 = t$ , then $x_1 = -5t$ and $x_2 = t$ . I believe this is the non-trivial linear dependency relationship, but I don't know if that the right way to leave it? Any help would be greatly appreciated.
September 9th 2009, 04:12 AM
BobP

If the vectors are linearly dependent there will be a set of coefficients, $\alpha, \beta, \gamma,$ not all zero, such that

$\alpha v_1 + \beta v_2 + \gamma v_3 = 0.$

This you can translate into a set of three linear equations.

For non-trivial solutions, it is necessary that the determinant of coefficients is zero, and this you have demonstrated, in which case the vectors are indeed linearly dependent.

Easier though is to simply solve the equations using elementary row operations. I get

$\alpha = -5t, \beta = t, \gamma = t,$

where $t$ is some arbitrary number.

(If the equations turn out to be inconsistent, it means that the vectors are linearly independent.)
September 10th 2009, 01:13 AM
drew.walker

So when it says "find a non-trivial linear dependency relationship", would this satisfactorily answer that part of the question:

$ \alpha = -5t, \beta = t, \gamma = t, $