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:

$\displaystyle

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:

$\displaystyle

\begin{bmatrix}

2 & 4 & 6 \\

1 & 4 & 1 \\

2 & 5 & 5

\end{bmatrix}

$

$\displaystyle

\begin{bmatrix}

1 & 0 & 5 \\

1 & 4 & 1 \\

2 & 5 & 5

\end{bmatrix}

$

$\displaystyle

\begin{bmatrix}

1 & 0 & 5 \\

1 & 4 & 1 \\

0 & -3 & 3

\end{bmatrix}

$

$\displaystyle

\begin{bmatrix}

1 & 0 & 5 \\

1 & 1 & 4 \\

0 & -3 & 3

\end{bmatrix}

$

$\displaystyle

\begin{bmatrix}

1 & 0 & 5 \\

0 & 1 & -1 \\

0 & -3 & 3

\end{bmatrix}

$

$\displaystyle

\begin{bmatrix}

1 & 0 & 5 \\

0 & 1 & -1 \\

0 & 0 & 0

\end{bmatrix}

$

If we let $\displaystyle x_4 = t$, then $\displaystyle x_1 = -5t$ and $\displaystyle x_2 = t$. 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.