Matrix invertibility proof?

Question

Can it be proven that $A^\top A$ is invertible given just the fact that:

if $A^\top Ay = \theta$ then $Ay = \theta$?

Here $y$ is a vector and $\theta$ is the vector zero.

Answer 1

No, this is not enough to show $A^T A$ is invertible. The claim holds when $A$ is the zero matrix (since $Ay=0$ is automatically true), yet in this case $A^T A$ is not invertible.

In fact the highlighted fact is true for every matrix, whether square or rectangular, singular or invertible (assuming we work over the reals and not some other field). If $A^T A y = 0$ then $y^T A^T A y = 0$, but the LHS is just $\| Ay \|$, so $Ay$ must be the zero vector.

Answer 2

If $A$ is injective then $A^TA$ is invertible. Maybe that's what you mean?