Is a set with two collinear vectors always linearly dependent?

Question

Say you have $\vec v_1 =\begin{bmatrix}1\\2\\3\end{bmatrix}$ $\vec v_2 =\begin{bmatrix}-1\\-2\\-3\end{bmatrix}$ and $\vec v_3 =$ anything.

Is the set $\{\vec v_1, \vec v_2, \vec v_3\}$ always considered linearly dependent because $\{\vec v_1, \vec v_2\}$ is?

Answer 1

The requirement for linear dependence is that $a_1v_1+a_2v_2+a_3v_3=0$, where not all the $a$s are zero. So if $a_1=a_2$ and $a_3=0$, then that would make it zero, so yes, they will always be linearly dependent

Answer 2

If you want to check by the definition:

$$1\cdot \vec{v_1}+1\cdot \vec{v_2} +0\cdot \vec{v_3} =\vec0$$

and not all the coefficients are zero, thus there is lin. dependence.