MathDB

Suppose that

V

is a finite dimensional vector space over the real numbers equipped with an inner product and

S:V\times V \longrightarrow \mathbb R

is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation

T:V \longrightarrow V

such that

\forall u,v \in V: S(u,v)=<u,T(v)>

.
We know that there always exists

v\in V

such that

W=<v,T(v)>

is invariant under

T

. (it means

T(W)\subseteq W

). Prove that if

W

is invariant under

T

then the following subspace is also invariant under

T

W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}

.
Prove that if dimension of

V

is more than

3

, then there exist a two dimensional subspace

W

V

such that the volume defined on it by function

S

is zero!!!!
(This is the way that we can define a two dimensional volume for each subspace

V

. This can be done for volumes of higher dimensions.)

Problem Statement