say $\mathbb{A}:V\rightarrow V$ is a linear and symmetric function with V an $\mathbb{R}$-vector space.
If $c=(c_1,...,c_n)$ is a orthonormal basis of $V$ (not spectral basis), $A$ the matrix of $\mathbb{A}$ regarding that basis (spectral theorem), $v=\sum_{i=1}^n v_i\cdot c_i, k=(v_1,...,v_n)\in\mathbb{R}^n$, then why is
$\mathbb{A}(v)=(Ak)^Tc=k^TA^Tc$?
Thank you!