im new here, and im having trouble checking some old theorems i didnt prove in their time. This is my first post, and also English is not my native language, so please be patient with me, i will update my question and fix any mistake or add any other requirements y may be missing. I currently only need to prove this, which as been a bit confusing:
Given $L$ a linear transform over $S$ subspace of $R^n$, and $B$ orthonormal base of $S$. Prove the matrix $M(L)$ representing the transform $L$ after a base change over $B$ (from and into) is also symmetric iff $L$ symmetric.
In other words:
$L$ symmetric <=> $M(L)$ symmetric
PD: i can't directly use $B^⊤$ = $B^{-1}$ if $B$ the orthonormal matrix generated by the base $B$.
If $B$ is the orthonormal matrix given by sending the standard basis to your orthonormal basis, then $B$ satisfies $B^{\top} = B^{-1}$, and in particular, if $M_B(L)$ denotes the transformed $L$ with matrix $B$, then $$ M_B(L)^{\top} = (BLB^{-1})^{\top} = (BLB^{\top})^{\top} = (B^{\top})^{\top} L^{\top} B^{\top} = BLB^{\top} = M_B(L). $$ Since $L = M_{B^{-1}}(M_B(L))$, you can repeat the argument by substituting $(B,L)$ by $(B^{-1}, M_B(L))$ to prove the equivalence.
Hope that helps,