So, I am trying to prove $$\|Mx\|^2 =x^TM^TMx,$$ however I am running into some difficulties.
Here, $M \in \mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$. I know that when you take the transpose you get a $1 \times n$ matrix for $x^T$ and an $n \times m$ matrix for $M^T$. So, when you multiply those you get a $1 \times m$ matrix, then again if you do that times $M$ you get a $1 \times n$ matrix, and then finally a $1 \times 1$ matrix, but I am a bit unsure on how to actually go about the proof. Any help would be appreciated.