Orthonormal basis and vectors in a space V

Question

Let's say I have an orthonormal basis of $V$ $(e_1,\ldots,e_n)$ and a bunch of vectors in $V$ $(v_1,\ldots,v_n)$ that may or may not be linearly independent. How can I then prove that those vectors form a basis of $V$, given that the difference in length squared between any vector $v_i$ and $e_i$ (which comes from the orthonormal basis) in $V$ is equal to $\frac{1}{\dim V}$? I can't use the Gram-Schmidt process to find an orthonormal basis because those vectors may not be linearly independent. I have also tried to take advantage of the inner product space property in expanding the length squared, but to no avail, because $v_i$ and $e_i$ are not necessarily orthogonal (and to make them orthogonal, one would have to use Gram-Schmidt, but that may not possible in this case because of the linear independence requirement). I am thinking that the relationship between $v_i$ and $e_i$ is the key component in this problem, but how do I make sense of it?