Math Problems Vault

Problem 9D.1

Suppose $v \in V$ and $w \in W$ . Prove that $v \otimes w = 0$ if and only if $v = 0$ or $w = 0$ .

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Problem 9D.2

Give an example of six distinct vectors $v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}$ in $\mathbb {R}^{3}$ such that

v_{1} \otimes w_{1}+v_{2} \otimes w_{2}+v_{3} \otimes w_{3} = 0

but none of $v_{1} \otimes w_{1}, v_{2} \otimes w_{2}, v_{3} \otimes w_{3}$ is a scalar multiple of another element of this list.

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Problem 9D.3

Suppose that $v_{1}, \ldots , v_{m}$ is a linearly independent list in $V$ . Suppose also that $w_{1}, \ldots , w_{m}$ is a list in $W$ such that

v_{1} \otimes w_{1}+\cdots +v_{m} \otimes w_{m} = 0.

Prove that $w_{1} = \cdots = w_{m} = 0$ .

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Problem 9D.4

Suppose $\operatorname {dim} V > 1$ and $\operatorname {dim} W > 1$ . Prove that

\left\{ v \otimes w:(v, w) \in V \times W\right\}

is not a subspace of $V \otimes W$ . This exercise implies that if $\operatorname {dim} V > 1$ and $\operatorname {dim} W > 1$ , then

\left\{ v \otimes w:(v, w) \in V \times W\right\} \neq V \otimes W.

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Problem 9D.5

Suppose $m$ and $n$ are positive integers. For $v \in \mathbf{F}^{m}$ and $w \in \mathbf{F}^{n}$ , identify $v \otimes w$ with an $m$ -by- $n$ matrix as in Example 9.76. With that identification, show that the set

\left\{ v \otimes w: v \in \mathbf{F}^{m} \text{ and } w \in \mathbf{F}^{n}\right\}

is the set of $m$ -by- $n$ matrices (with entries in $\mathbf{F}$ ) that have rank at most one.

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Remark.

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Example 9.76: let $e_{1}, \ldots , e_{m}$ denote the standard basis of $\mathbf{F}^{m}$ and let $f_{1}, \ldots , f_{n}$ denote the standard basis of $\mathbf{F}^{n}$ . Suppose $v = (v_{1}, \ldots , v_{m}) \in \mathbf{F}^{m}$ and $w = (w_{1}, \ldots , w_{n}) \in \mathbf{F}^{n}$ . Then $v \otimes w = \left(\sum_{j=1}^{m} v_{j} e_{j}\right) \otimes \left(\sum_{k=1}^{n} w_{k} f_{k}\right) = \sum_{k=1}^{n}\sum_{j=1}^{m} (v_{j} w_{k})(e_{j} \otimes f_{k})$ . Writing the coefficients $v_{j}w_{k}$ in an $m$ -by- $n$ matrix with $v_{j}w_{k}$ in row $j$ , column $k$ , identifies $v \otimes w$ with the $m$ -by- $n$ matrix whose $(j,k)$ entry is $v_{j}w_{k}$ .

Problem 9D.6

Suppose $m$ and $n$ are positive integers. Give a description, analogous to Exercise 5, of the set of $m$ -by- $n$ matrices (with entries in $\mathbf{F}$ ) that have rank at most two.

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Problem 9D.7

Suppose $\operatorname {dim} V > 2$ and $\operatorname {dim} W > 2$ . Prove that

\left\{ v_{1} \otimes w_{1}+v_{2} \otimes w_{2}: v_{1}, v_{2} \in V \text{ and } w_{1}, w_{2} \in W\right\} \neq V \otimes W

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Problem 9D.8

Suppose $v_{1}, \ldots , v_{m} \in V$ and $w_{1}, \ldots , w_{m} \in W$ are such that

v_{1} \otimes w_{1}+\cdots +v_{m} \otimes w_{m} = 0

Suppose that $U$ is a vector space and $\Gamma : V \times W \rightarrow U$ is a bilinear map. Show that

\Gamma \left(v_{1}, w_{1}\right)+\cdots +\Gamma \left(v_{m}, w_{m}\right) = 0

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Problem 9D.9

Suppose $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$ . Prove that there exists a unique operator on $V \otimes W$ that takes $v \otimes w$ to $S v \otimes T w$ for all $v \in V$ and $w \in W$ .

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Remark.

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In an abuse of notation, the operator on $V \otimes W$ given by this exercise is often called $S \otimes T$ .

Problem 9D.10

Suppose $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$ . Prove that $S \otimes T$ is an invertible operator on $V \otimes W$ if and only if both $S$ and $T$ are invertible operators. Also, prove that if both $S$ and $T$ are invertible operators, then $(S \otimes T)^{-1} = S^{-1} \otimes T^{-1}$ , where we are using the notation from the comment after Exercise 9.

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Remark.

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The comment after Exercise 9: in an abuse of notation, the operator on $V \otimes W$ that takes $v \otimes w$ to $Sv \otimes Tw$ for all $v \in V$ and $w \in W$ (given by Exercise 9) is often called $S \otimes T$ .

Problem 9D.11

Suppose $V$ and $W$ are inner product spaces. Prove that if $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$ , then $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ , where we are using the notation from the comment after Exercise 9.

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Remark.

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The comment after Exercise 9: in an abuse of notation, the operator on $V \otimes W$ that takes $v \otimes w$ to $Sv \otimes Tw$ for all $v \in V$ and $w \in W$ (given by Exercise 9) is often called $S \otimes T$ .

Problem 9D.12

Suppose that $V_{1}, \ldots , V_{m}$ are finite-dimensional inner product spaces. Prove that there is a unique inner product on $V_{1} \otimes \cdots \otimes V_{m}$ such that

\left\langle v_{1} \otimes \cdots \otimes v_{m}, u_{1} \otimes \cdots \otimes u_{m} \right\rangle = \left\langle v_{1}, u_{1} \right\rangle \cdots \left\langle v_{m}, u_{m} \right\rangle

for all $\left(v_{1}, \ldots , v_{m}\right)$ and $\left(u_{1}, \ldots , u_{m}\right)$ in $V_{1} \times \cdots \times V_{m}$ . Note that the equation above implies that

\left\| v_{1} \otimes \cdots \otimes v_{m}\right\| = \left\| v_{1}\right\| \times \cdots \times \left\| v_{m}\right\|

for all $\left(v_{1}, \ldots , v_{m}\right) \in V_{1} \times \cdots \times V_{m}$ .

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Problem 9D.13

Suppose that $V_{1}, \ldots , V_{m}$ are finite-dimensional inner product spaces and $V_{1} \otimes \cdots \otimes V_{m}$ is made into an inner product space using the inner product from Exercise 12. Suppose $e_{1}^{k}, \ldots , e_{n_{k}}^{k}$ is an orthonormal basis of $V_{k}$ for each $k = 1, \ldots , m$ . Show that the list

\left\{ e_{j_{1}}^{1} \otimes \cdots \otimes e_{j_{m}}^{m}\right\} _{j_{1} = 1, \ldots , n_{1} ; \cdots ; j_{m} = 1, \ldots , n_{m}}

is an orthonormal basis of $V_{1} \otimes \cdots \otimes V_{m}$ .

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Remark.

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Exercise 12: the inner product on $V_{1} \otimes \cdots \otimes V_{m}$ is the unique one such that $\left\langle v_{1} \otimes \cdots \otimes v_{m}, u_{1} \otimes \cdots \otimes u_{m} \right\rangle = \left\langle v_{1}, u_{1} \right\rangle \cdots \left\langle v_{m}, u_{m} \right\rangle$ for all $\left(v_{1}, \ldots , v_{m}\right)$ and $\left(u_{1}, \ldots , u_{m}\right)$ in $V_{1} \times \cdots \times V_{m}$ .