In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. V.A.1) Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal. This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$. V.A.2) If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?
In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
\textbf{V.A.1)} Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal.
This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$.
\textbf{V.A.2)} If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?