Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is: $$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$ This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$. b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$.
a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is:
$$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$
This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$.
b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.