Answer
False
Work Step by Step
Consider the vectors $(1,0)$ and $(1,1)$ in $R^{ \ 2}$ with the standard inner product.
We have that $(1,0)$ and $(1,1)$ linearly independent in $R^{\ 2}$ since they are not
proportional and $\langle(1,0),(1,1)\rangle= 1 \cdot 1+0 \cdot 1=1+0=1$
Thus $\{(1,0),(1,1)\}$ is a linearly independent set in $\mathbb{R}^{2}$ which is not orthogonal.