grandes-ecoles 2020 Q7

grandes-ecoles · France · mines-ponts-maths1__mp_cpge Matrices Linear Transformation and Endomorphism Properties

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:
$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$
Let $a \in E$ and $x \in E \backslash \{0\}$. Show that $\operatorname{tr}(a \otimes x) = (a \mid x)$.

We consider a Euclidean vector space $(E, (-\mid-))$. Given $a \in E$ and $x \in E$, we denote by $a \otimes x$ the map from $E$ to itself defined by:

$$\forall z \in E, (a \otimes x)(z) = (a \mid z) \cdot x$$

Let $a \in E$ and $x \in E \backslash \{0\}$. Show that $\operatorname{tr}(a \otimes x) = (a \mid x)$.

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