grandes-ecoles 2010 QII.A.2

grandes-ecoles · France · centrale-maths2__mp Proof Direct Proof of a Stated Identity or Equality

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.
Let $F$ and $G$ be two vector subspaces of $E$.
a) Show that $(F+G)^\perp = F^\perp \cap G^\perp$.
b) Show that $(F \cap G)^\perp = F^\perp + G^\perp$.

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.

Let $F$ and $G$ be two vector subspaces of $E$.

a) Show that $(F+G)^\perp = F^\perp \cap G^\perp$.

b) Show that $(F \cap G)^\perp = F^\perp + G^\perp$.

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