Given that $A \in \mathbb{R}^{m \times n}$, $\vec{x} \in \mathbb{R}^n$, $\vec{b} \in \mathbb{R}^m$, $\vec{b} \neq \vec{0}$ and the equation $A \vec{x}= \vec{b}$ only has one solution, how do we prove/disprove the following statements?
1.
$$\|A \vec{x}\| = \|A^TA \vec{x}\|$$ 2. $$\|A \vec{x}\| = \|A(A^TA)^{-1}A^T \ \vec{b}\|$$