Chapter 14

Show that if \(\tau: W \rightarrow X\) is a linear map and \(b: U \times V \rightarrow W\) is bilinear, then \(\tau \circ b: U \times V \rightarrow X\) is bilinear.

Let \(\mathcal{B}=\left\\{u_{i} \mid i \in I\right\\}\) be a basis for \(U\) and let \(\mathcal{C}=\left\\{v_{j} \mid j \in J\right\\}\) be a basis for \(V\). Show that the set $$ \mathcal{D}=\left\\{u_{i} \otimes v_{j} \mid i \in I, j \in J\right\\} $$ is a basis for \(U \otimes V\) by showing that it is linearly independent and spans.

Prove that the following property of a pair \((W, g: U \times V \rightarrow W)\) with \(g\) bilinear characterizes the tensor product \((U \otimes V, t: U \times V \rightarrow U \otimes V)\) up to isomorphism, and thus could have been used as the definition of tensor product: For a pair \((W, g: U \times V \rightarrow W)\) with \(g\) bilinear if \(\left\\{u_{i}\right\\}\) is a basis for \(U\) and \(\left\\{v_{i}\right\\}\) is a basis for \(V\), then \(\left\\{g\left(u_{i}, v_{j}\right)\right\\}\) is a basis for \(W\).

Prove that \(U \otimes V \approx V \otimes U\).

Let \(\mathcal{B}=\left\\{b_{i}\right\\}\) be a basis for \(U\) and \(\mathcal{C}=\left\\{c_{i}\right\\}\) be a basis for \(V\). Show that any function \(f: \mathcal{B} \times \mathcal{C} \rightarrow W\) can be extended to a linear function \(\bar{f}: U \otimes V \rightarrow W\). Deduce that the function \(f\) can be extended in a unique way to a bilinear map \(\widehat{f}: U \times V \rightarrow W\). Show that all bilinear maps are obtained in this way.

Let \(S \subseteq U\) and \(T \subseteq V\) be subspaces of vector spaces \(U\) and \(V\), respectively. Show that $$ (S \otimes V) \cap(U \otimes T) \approx S \otimes T $$

Let \(S_{1}, S_{2} \subseteq U\) and \(T_{1}, T_{2} \subseteq V\) be subspaces of \(U\) and \(V\), respectively. Show that $$ \left(S_{1} \otimes T_{1}\right) \cap\left(S_{2} \otimes T_{2}\right) \approx\left(S_{1} \cap S_{2}\right) \otimes\left(T_{1} \otimes T_{2}\right) $$

Find an example of two vector spaces \(U\) and \(V\) and a nonzero vector \(x \in U \otimes V\) that has at least two distinct (not including order of the terms) representations of the form $$ x=\sum_{i=1}^{n} u_{i} \otimes v_{i} $$ where the \(u_{i}\) 's are linearly independent and so are the \(v_{i}^{\prime}\) s.

Let \(\iota_{X}\) denote the identity operator on a vector space \(X\). Prove that \(\iota_{V} \odot \iota_{W}=\iota_{V \otimes W}\).

Suppose that \(\tau_{1}: U \rightarrow V, \tau_{2}: V \rightarrow W\) and \(\sigma_{1}: U^{\prime} \rightarrow V_{K}, \sigma_{2}: V_{K} \rightarrow W^{\prime}\). Prove that $$ \left(\tau_{2} \circ \tau_{1}\right) \odot\left(\sigma_{2} \circ \sigma_{1}\right)=\left(\tau_{2} \odot \sigma_{2}\right) \circ\left(\tau_{1} \odot \sigma_{1}\right) $$

Connect the two approaches to extending the base field of an \(F\)-space \(V\) to \(K\) (at least in the finite-dimensional case) by showing that \(F^{n} \otimes_{F} K \approx(K)^{n}\).

Prove that in a tensor product \(U \otimes U\) for which \(\operatorname{dim}(U) \geq 2\) not all vectors have the form \(u \otimes v\) for some \(u, v \in U\). Hint. Suppose that \(u, v \in U\) are linearly independent and consider \(u \otimes v+v \otimes u\).

Prove that for the block matrix $$ M=\left[\begin{array}{ll} A & B \\ 0 & C \end{array}\right]_{\text {block }} $$ we have \(d(M)=d(A) d(C)\).

Let \(A, B \in M_{n}(F)\). Prove that if either \(A\) or \(B\) is invertible, then the matrices \(A+\alpha B\) are invertible except for a finite number of \(\alpha\) 's. The Tensor Product of Matrices

Let \(A=\left(a_{i, j}\right)\) be the matrix of a linear operator \(\tau \in \mathcal{L}(V)\) with respect to the ordered basis \(\mathcal{A}=\left(u_{1}, \ldots, u_{n}\right)\). Let \(B=\left(b_{i, j}\right)\) be the matrix of a linear operator \(\sigma \in \mathcal{L}(V)\) with respect to the ordered basis \(\mathcal{B}=\left(v_{1}, \ldots, v_{m}\right)\). Consider the ordered basis \(\mathcal{C}=\left(u_{i} \otimes v_{j}\right)\) ordered lexicographically; that is \(u_{i} \otimes v_{j}

Show that the tensor product is not, in general, commutative.

Show that the tensor product \(A \otimes B\) is bilinear in both \(A\) and \(B\).

Show that \(A \otimes B=0\) if and only if \(A=0\) or \(B=0\).

Show that a) \((A \otimes B)^{t}=A^{t} \otimes B^{t}\) b) \((A \otimes B)^{*}=A^{*} \otimes B^{*}(\) when \(F=\mathbb{C})\)

Suppose that \(A_{m, n}, B_{p, q}, C_{n, k}\) and \(D_{q, r}\) are matrices of the given sizes. Prove that $$ (A \otimes B)(C \otimes D)=(A C) \otimes(B D) $$ Discuss the case \(k=r=1\).

Prove that if \(A\) and \(B\) are nonsingular, then so is \(A \otimes B\) and $$ (A \otimes B)^{-1}=A^{-1} \otimes B^{-1} $$

Prove that \(\operatorname{tr}(A \otimes B)=\operatorname{tr}(A) \cdot \operatorname{tr}(B)\).

Suppose that \(F\) is algebraically closed. Prove that if \(A\) has eigenvalues \(\lambda_{1}, \ldots, \lambda_{n}\) and \(B\) has eigenvalues \(\mu_{1}, \ldots, \mu_{m}\), both lists including multiplicity, then \(A \otimes B\) has eigenvalues \(\left\\{\lambda_{i} \mu_{j} \mid i \leq n, j \leq m\right\\}\), again counting multiplicity.

Prove that \(\operatorname{det}\left(A_{n, n} \otimes B_{m, m}\right)=\left(\operatorname{det}\left(A_{n, n}\right)\right)^{m}\left(\operatorname{det}\left(B_{m, m}\right)\right)^{n}\).