grandes-ecoles 2014 QIII.B.3

grandes-ecoles · France · centrale-maths2__pc Matrices Linear Transformation and Endomorphism Properties

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Prove that $n$ is an element of $\{1, 2, 4, 8\}$.

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that
$$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$
We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$.\\
Prove that $n$ is an element of $\{1, 2, 4, 8\}$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.E.1 QI.E.2 QI.E.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4