Clu Linear Algebra Ch 1 Example Problems

Example Question

Are the three vectors that make up the following augmented matrix linearly independent?

(1)

\begin{matrix} 1 & 1 & 3 & | & 0 \\ -1 & 3 & 1 & | & 0 \\ 2 & -1 & 3 & | & 0 \\ \end{matrix}

No, because the matrix row reduces to

(2)

\begin{matrix} 1 & 1 & 3 & | & 0 \\ 0 & 1 & 1 & | & 0 \\ 0 & 0 & 0 & | & 0 \\ \end{matrix}

which has a free variable ($x_3$). This means all the row equations will be in terms of $x_3$. Therefore, the three vectors are linearly DEPENDENT