Subspace generated by a family

Matrix.
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5 \\ 3 & 5 & 7 & 9 \end{pmatrix}.$$

Row reduction. $L_2\leftarrow L_2-2L_1$, $L_3\leftarrow L_3-3L_1$:
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & -1 & -2 & -3 \\ 0 & -1 & -2 & -3 \end{pmatrix}.$$
$L_3\leftarrow L_3-L_2$:
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & -1 & -2 & -3 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$$
Rank is $2$.

Conclusion. $\dim\operatorname{span}(v_1,v_2,v_3)=2$. Note that $v_3=v_1+v_2$, so the family is dependent.