grandes-ecoles 2010 QII.B.2

grandes-ecoles · France · centrale-maths2__mp Proof True/False Justification

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.
With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?

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.

With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?

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