One of my lectures includes the following quote from my professor (on a part of a chapter about compatible systems):
$A^TA$ is semi-positive-definite. If columns of $A$ are linear independent, then $A^TA$ is positive definite and invertible and has only one solution.
I'm fine with the first part of it that it's positive semi-definite since
$\forall x(\neq 0): X^T(A^TA)X=\big\|AX\big\|_2^2 \geq0$
I don't get how from its columns being linearly independent we get to it's being PD.