Answer
False
Work Step by Step
Consider the vectors $u=(1,0)$ and $v=(1,1)$ in $R^{\ 2}$ with the standard inner product.
Since $$\left|\begin{array}{ll}{1} & {1} \\ {0} & {1}\end{array}\right|=1-0=1 \neq 0$$
we get $u$ and $v$ are linearly independent.
We have also that $$\langle u, v\rangle= 1 \cdot 1+0 \cdot 1=1+0=1 \neq 0$$
So, the statement is False