Unique minimizer in a strictly convex function

Question

Consider the following minimization problem: $$ \min_x f(x):=(A^Tx -y)^2 $$ where $A$ is a vector and $y$ is a constant.

I am very confused with the following two contradicting conclusions:

1. We can find that $\nabla^2f(x) = AA^T >0$ if $A$ has all nonzero elements. It implies the function is strictly convex and there is an unique minimizer.
2. But obviously, there are infinite solutions as long as x satisfies $A^Tx = y$.

The question might be silly...But I need help to understand it.