Multilinear and alternating forms

Let $\omega^1,\ldots ,\omega^k$ be one-forms. Show that $\{ \omega^i\}_{i=1}^k$ are linearly independent if and only if $\omega^1 \wedge \omega^2 \wedge \cdots \wedge \omega^k \neq 0$.

Prove formula (46).

Formula (46) of Section 10.14: every $k$-tuple $\left\{ j_{1}, \ldots , j_{k}\right\}$ of distinct integers $1 \leq j_{i} \leq n$ can be converted to an increasing $k$-index $J$ by a finite number of interchanges of pairs, each of which amounts to a multiplication by $-1$; hence

where $\varepsilon \left(j_{1}, \ldots , j_{k}\right)$ is $1$ or $-1$, depending on the number of interchanges that are needed. Formula (46) asserts that

where $s$ is as in Definition 9.33: for an ordered $k$-tuple of integers, $s\left(j_{1}, \ldots , j_{k}\right) = \prod_{p<q} \operatorname {sgn}\left(j_{q}-j_{p}\right)$, with $\operatorname {sgn} x = 1$ if $x > 0$, $\operatorname {sgn} x = -1$ if $x < 0$, $\operatorname {sgn} x = 0$ if $x = 0$. (Here, for an increasing $k$-index $I = \left\{ i_{1}, \ldots , i_{k}\right\}$, $1 \leq i_{1} < \cdots < i_{k} \leq n$, one writes $d x_{I} = d x_{i_{1}} \wedge \cdots \wedge d x_{i_{k}}$.)

If $\omega$ and $\lambda$ are $k$ - and $m$-forms, respectively, prove that

Find formulas for $\operatorname {dim} V_{\mathrm{sym}}^{(2)}$ and $\operatorname {dim} V_{\mathrm{alt}}^{(2)}$ in terms of $\operatorname {dim} V$.

Suppose that $n$ is a positive integer and $V = \left\{ p \in \mathcal{P}_{n}(\mathbb {R}): p(0) = p(1)\right\}$. Define $\alpha : V \times V \rightarrow \mathbb {R}$ by

Show that $\alpha$ is an alternating bilinear form on $V$.

Suppose $m$ is a positive integer. Show that $\operatorname {dim} V^{(m)} = (\operatorname {dim} V)^{m}$.

Suppose $n \geq 3$ and $\alpha : \mathbf{F}^{n} \times \mathbf{F}^{n} \times \mathbf{F}^{n} \rightarrow \mathbf{F}$ is defined by

Show that $\alpha$ is an alternating 3-linear form on $\mathbf{F}^{n}$.

Suppose $m$ is a positive integer and $\alpha$ is an $m$-linear form on $V$ such that $\alpha \left(v_{1}, \ldots , v_{m}\right) = 0$ whenever $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$ with $v_{j} = v_{j+1}$ for some $j \in \left\{ 1, \ldots , m-1\right\}$. Prove that $\alpha$ is an alternating $m$-linear form on $V$.

Prove or give a counterexample: If $\alpha \in V_{\text{alt }}^{(4)}$, then

is a subspace of $V^{4}$.

Suppose $m$ is a positive integer and $\beta$ is an $m$-linear form on $V$. Define an $m$-linear form $\alpha$ on $V$ by

for $v_{1}, \ldots , v_{m} \in V$. Explain why $\alpha \in V_{\text{alt }}^{(m)}$.

Suppose $m$ is a positive integer and $\beta$ is an $m$-linear form on $V$. Define an $m$-linear form $\alpha$ on $V$ by

for $v_{1}, \ldots , v_{m} \in V$. Explain why

for all $v_{1}, \ldots , v_{m} \in V$ and all $\left(k_{1}, \ldots , k_{m}\right) \in \operatorname {perm} m$.

Give an example of a nonzero alternating 2-linear form $\alpha$ on $\mathbb {R}^{3}$ and a linearly independent list $v_{1}, v_{2}$ in $\mathbb {R}^{3}$ such that $\alpha \left(v_{1}, v_{2}\right) = 0$.