§ 3

A function of two vectors in a three-dimensional Euclidean vector space assigns to any two vectors $\mathbf{x}$ and $\mathbf{y}$ the scalar triple product $(\mathbf{a}, \mathbf{x}, \mathbf{y})$, where $\mathbf{a}$ is a fixed vector. Prove that this function is a 2-form. Express its matrix in a given basis $\mathbf{e}$ through the coordinates of the vector $\mathbf{a}$.

Let $x_{1}, \ldots , x_{n}$ be linearly independent vectors in an $n$-dimensional linear space $\mathcal{L}$. Prove that the bivectors $x_{k} \wedge x_{l}$ for all $k<l$ form a basis in the space of bivectors of the space $\mathcal{L}$.

Let $x_{1}, \ldots , x_{p}$ be a basis of a subspace $\mathcal{L}^{\prime } \subset \mathcal{L}$. Let us call $w=x_{1} \wedge \ldots \wedge x_{p}$ the direction $p$-vector of the subspace.

Prove that

Prove that in a three-dimensional space every bivector is simple.