grandes-ecoles 2010 QII.B.1

grandes-ecoles · France · centrale-maths2__mp Proof Computation of a Limit, Value, or Explicit Formula

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.
Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.

We assume that $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.

Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4