Math Problems Vault

Problem 9B.1

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

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Problem 9B.2

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

\begin{aligned} & \alpha \left(\left(x_{1}, \ldots , x_{n}\right),\left(y_{1}, \ldots , y_{n}\right),\left(z_{1}, \ldots , z_{n}\right)\right) \\ & \quad = x_{1} y_{2} z_{3}-x_{2} y_{1} z_{3}-x_{3} y_{2} z_{1}-x_{1} y_{3} z_{2}+x_{3} y_{1} z_{2}+x_{2} y_{3} z_{1}. \end{aligned}

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

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Problem 9B.3

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 $V$ .

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Problem 9B.4

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

\left\{ \left(v_{1}, v_{2}, v_{3}, v_{4}\right) \in V^{4}: \alpha \left(v_{1}, v_{2}, v_{3}, v_{4}\right) = 0\right\}

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

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Problem 9B.5

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

\alpha \left(v_{1}, \ldots , v_{m}\right) = \sum _{\left(j_{1}, \ldots , j_{m}\right) \in \operatorname {perm} m}\left(\operatorname {sign}\left(j_{1}, \ldots , j_{m}\right)\right) \beta \left(v_{j_{1}}, \ldots , v_{j_{m}}\right)

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

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Problem 9B.6

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

\alpha \left(v_{1}, \ldots , v_{m}\right) = \sum _{\left(j_{1}, \ldots , j_{m}\right) \in \operatorname {perm} m} \beta \left(v_{j_{1}}, \ldots , v_{j_{m}}\right)

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

\alpha \left(v_{k_{1}}, \ldots , v_{k_{m}}\right) = \alpha \left(v_{1}, \ldots , v_{m}\right)

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

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Problem 9B.7

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 $\alpha \left(v_{1}, v_{2}\right) = 0$ .

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Remark.

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This exercise shows that 9.39 can fail if the hypothesis that $n = \operatorname {dim} V$ is deleted. Theorem 9.39 (alternating $(\operatorname {dim} V)$ -linear forms and linear independence): let $n = \operatorname {dim} V$ . Suppose $\alpha$ is a nonzero alternating $n$ -linear form on $V$ and $e_{1}, \ldots , e_{n}$ is a list of vectors in $V$ . Then $\alpha (e_{1}, \ldots , e_{n}) \neq 0$ if and only if $e_{1}, \ldots , e_{n}$ is linearly independent.