Why $\left\langle y, A \varepsilon \right\rangle = \left\langle A^\intercal y, \varepsilon \right\rangle$ holds

Question

Why does $\left\langle y, A \varepsilon \right\rangle = \left\langle A^\intercal y, \varepsilon \right\rangle$ hold?

Given that $y \in \mathbb{R}^{n}, \varepsilon \in \mathbb{R}^{d}, A \in \mathbb{R}^{n \times d}$, and $\langle \cdot, \cdot \rangle$ is the dot product.

Answer 1

Use the definition of the dot product $\langle u,v\rangle=v^Tu$. Then we have

$$\langle y,A\varepsilon\rangle = (A\varepsilon)^T y=\varepsilon^T A^T y=\langle A^Ty,\varepsilon\rangle \,.$$