grandes-ecoles 2014 Q2

grandes-ecoles · France · x-ens-maths1__mp Matrices Bilinear and Symplectic Form Properties

Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.

Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20