My professor always writes on the board:
$A$ is $m \times n$, assuming that the vectors of $A$ form a basis, then $A^TA$ is always invertible.
one thing I know is that $A^TA$ is always symmetric, but I'm not sure about the conditions on a symmetric matrix needed to ensure that it is invertible?
@RobertLewis
A Gram matrix is usually defined by giving a set of vectors and then defining the i,j entry as the dot product of the i,j vectors. In doing so, clearly the set of vectors can be thought of as column vectors of A. So saying "the vectors for A" is a completely natural thing to say, and should be unambiguous.
here is an elegant proof Gram matrix invertible iff set of vectors linearly independent