Answer
see the proof below.
Work Step by Step
Let $A$ be an $n\times n $ matrix and $u,v\in R^n$, then we have
(a) \begin{aligned}
\left\langle A^{T} u, v\right\rangle &=\left(A^{T} u\right)^{T} v \\ &=u^{T} A v \\ &=u^{T}(A v) \\ &=\langle u, A v\rangle
\end{aligned}.
(b)
\begin{aligned}\left\langle A^{T} A u, u\right\rangle &=\left(A^{T} A u\right)^{T} u \\ &=u^{T} A^{T} A u \\ &=(A u)^{T} A u \\ &=\langle A u, A u\rangle \\ &=\|A u\|^{2} .\end{aligned}