grandes-ecoles 2010 QII.A.4

grandes-ecoles · France · centrale-maths2__mp Proof Proof of Set Membership, Containment, or Structural Property

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 say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.
If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.

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 say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.

If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.

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