Answer
See explanation
Work Step by Step
Let:
$\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ is a linearly independent set in $\mathbb{R}^{n}$
Goal:
Explain $\left\{\mathbf{v}_{1}, \mathbf{v}_{1}+\mathbf{v}_{2}\right\}$ is also linearly independent.
Concepts:
Linear Independence
Solve
Let $c_{1}$ and $c_{2}$ are constants such that
\[
c_{1} \mathbf{v}_{1}+c_{2}\left(\mathbf{v}_{1}+\mathbf{v}_{2}\right)=\mathbf{0}
\]
Then, $\left(c_{1}+c_{2}\right) \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}=\mathbf{0}$
$\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are linearly independent, so $c_{1}+$ $c_{2}=0$ and $c_{2}=0$
Result
Thus both $c_{1}$ and $c_{2}$ in equation (1) must also be
zero.
Conclusion
$\left\{\mathbf{v}_{1}, \mathbf{v}_{1}+\mathbf{v}_{2}\right\}$ is linearly independent.
