Properties of transpose

Let $A$ be a real $n \times n$ matrix and let $A^{⊤}$ denote its transpose.

Prove that $(A v) \cdot w = v \cdot (A^{⊤} w)$ for all vectors $v, w \in R^{n}$ . Hint: Recall that the dot product $u \cdot v$ equals the matrix product $u^{⊤} v$ .
Suppose now $A$ is also symmetric, i.e., that $A^{⊤} = A$ . Also suppose $v$ and $w$ are eigenvectors of $A$ with different eigenvalues. Prove that $v$ and $w$ are orthogonal.

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Let $A$ be a real $n\times n$ matrix and let $A^{\top}$ denote its transpose.
\begin{enumerate}[label=\alph*)]
	\item Prove that $(A{\bf v})\cdot {\bf w}= {\bf v}\cdot (A^{\top}{\bf w})$ for all vectors ${\bf v},{\bf w}\in {\bf R}^n$. {\itshape Hint:} Recall that the dot product ${\bf u}\cdot {\bf v}$ equals the matrix product ${\bf u}^{\top}{\bf v}$.
	\item Suppose now $A$ is also symmetric, i.e., that $A^{\top} = A$. Also suppose ${\bf v}$ and ${\bf w}$ are eigenvectors of $A$ with different eigenvalues. Prove that ${\bf v}$ and ${\bf w}$ are orthogonal.
\end{enumerate}