Consider some linearly independent vectors $\mathbf{\bar{v}}_1, \mathbf{\bar{v}}_2, \dots, \mathbf{\bar{v}}_m$ in $\mathbb{R}^n$ and a vector $\mathbf{\bar{v}} \in \mathbb{R}^n$ that is not contained in the span of $\mathbf{\bar{v}}_1, \mathbf{\bar{v}}_2, \dots, \mathbf{\bar{v}}_m$.
Are the vectors $\mathbf{\bar{v}}_1, \mathbf{\bar{v}}_2, \dots, \mathbf{\bar{v}}_m, \mathbf{\bar{v}}$ necessarily linearly independent?
Yes, because if they are not, we have scalars $\lambda_i$, not all zero, such that $$ \sum_i \lambda_i v_i + \lambda v =0; $$ further, since the $v_i$ are linearly independent, $\lambda \neq 0$. But then $$ v = \sum_i \frac{\lambda_i}{\lambda}v_i, $$ so by the definition of span, $v$ is in the span of the $v_i$. Contradiction. Hence $\{ v_i \}_i \cup \{ v \}$ is a linearly independent set.