Gram matrix invertible iff set of vectors linearly independent Given a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as
$G_{i,j}=v_i \cdot v_j$
Due to symmetry in the dot product, $G$ is Hermitian.
I'm trying to remember why $|G|=0$ iff the set of vectors are not linearly independent.

For one thing, the Gram determinant is the square of the $n$-dimensional volume of the parallelopiped generated by the given basis vectors. The determinant is 0 when that object is not full dimensional, that is when one of the vectors lies in the hyperplane generated by the other $n-1$ vectors, in turn meaning linear dependence.
