## Question

###### Let $V$ be a vector space over $K$. A mapping $f: \widehat{V \times V \times \ldots \times V} \rightarrow K$ is called a multilinear (or $m$ -linear) form on $V$ if $f$ is linear in each variable; that is, for $i=1, \ldots, m$ \[ f(\ldots, \widehat{a u+b v}, \ldots)=a f(\ldots, \hat{u}, \ldots)+b f(\ldots, \hat{v}, \ldots) \] where $\widehat{\longrightarrow}$ denotes the $i$ th element, and other elements are held fixed. An $m$ -linear form $f$ is said to be alternating if $f\left(v_{1}, \ldots

Let $V$ be a vector space over $K$. A mapping $f: \widehat{V \times V \times \ldots \times V} \rightarrow K$ is called a multilinear (or $m$ -linear) form on $V$ if $f$ is linear in each variable; that is, for $i=1, \ldots, m$ \[ f(\ldots, \widehat{a u+b v}, \ldots)=a f(\ldots, \hat{u}, \ldots)+b f(\ldots, \hat{v}, \ldots) \] where $\widehat{\longrightarrow}$ denotes the $i$ th element, and other elements are held fixed. An $m$ -linear form $f$ is said to be alternating if $f\left(v_{1}, \ldots v_{m}\right)=0$ whenever $v_{i}=v_{j}$ for $i \neq j .$ Prove the following: (a) The set $B_{m}(V)$ of $m$ -linear forms on $V$ is a subspace of the vector space of functions from $V \times V \times \cdots \times V$ into $K$ (b) The set $A_{m}(V)$ of alternating $m$ -linear forms on $V$ is a subspace of $B_{m}(V)$ Remark 1: If $m=2,$ then we obtain the space $B(V)$ investigated in this chapter. Remark 2: If $V=K^{m}$, then the determinant function is an alternating $m$ -linear form on $V$