Tensor Products of Spaces

Verify that $V \otimes W$ is a linear space.

Let $\mathbb {k}[x]$ be the space of all polynomials in the variable $x$ with coefficients in the field $\mathbb {k}$, and let $\mathbb {k}[x, y]$ be the space of polynomials in $x$ and $y$. Prove that the correspondence $f \otimes g \mapsto f g$ defines a linear isomorphism $\mathbb {k}[x] \otimes \mathbb {k}[y] \rightarrow \mathbb {k}[x, y]$.

Let $\mathbb {k}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with coefficients in the field $\mathbb {k}$, and let $\mathbb {k}_{n, m}[x, y]$ be the space of polynomials in $x$ and $y$ of degree at most $n$ in $x$ and at most $m$ in $y$. Prove that the correspondence $f \otimes g \mapsto f g$ defines a linear isomorphism $\mathbb {k}_{n}[x] \otimes \mathbb {k}_{m}[y] \rightarrow \mathbb {k}_{n, m}[x, y]$.

Prove the associativity and commutativity (up to natural isomorphisms) of the tensor product operation on spaces.

Establish the natural isomorphism

Consider $F \in \mathbf{P}_{q+q^{\prime }}^{p+p^{\prime }}$. Prove that the map

defines a linear map $\widetilde{F}: \mathbf{P}_{p}^{q} \rightarrow \mathbf{P}_{q^{\prime }}^{p^{\prime }}$.

Let $\operatorname {dim} V = n, \operatorname {dim} W = m$, and let $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{m}$ be some bases of the spaces $V$ and $W$. Prove that $\operatorname {dim}(V \otimes W) = n \cdot m$, and that $e_{i} \otimes f_{j}, i = 1, \ldots , n, j = 1, \ldots , m$ can be chosen as a basis.

Let there be given decompositions of spaces into direct sums: $V = \bigoplus_{i} V_{i}, W = \bigoplus_{j} W_{j}$. Prove that $V \otimes W \cong \bigoplus_{i, j} V_{i} \otimes W_{j}$ (after the natural identification).

In a special case, the isomorphism between the tensor product and the space of tensors (see problem 1706) becomes the canonical isomorphism $\tau : V^{*} \otimes V \rightarrow \mathscr {L}(V)$, where $\mathscr {L}(V)$ is the space of linear operators. Compute (first justifying independence of the dimension):

In the notation of the previous problem, find, for $\operatorname {dim} V = 4$

where $\circ$ denotes composition of maps.

Let $V = \mathbb {R}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with real coefficients, and let the function $l: V \rightarrow \mathbb {R}$, given by the formula

define a covector. Also consider the polynomials $r = 1-x+x^{2}$ and $q = 1+x+x^{2}+\ldots +x^{n}$. Find $\tau (l \otimes r)(q)$.

Show that $f \otimes g$, as defined in the theoretical part, is a well-defined linear map.

Let us denote $V^{\otimes p} = \underbrace{V \otimes \ldots \otimes V}_{p}$ and $f^{\otimes p} = \underbrace{f \otimes \ldots \otimes f}_{p}$, where $f$ is a linear map. Prove that $\operatorname {rk}\left(f^{\otimes p}\right) = (\operatorname {rk}\left(f\right))^{p}$. In particular, the tensor power of an invertible operator is invertible.

For operators in the space $V$, prove the formulas:

Prove that $\operatorname {tr}\left(f^{\otimes p}\right) = (\operatorname {tr}(f))^{p}$, where $f$ is a linear operator.

Let $e_{1}, \ldots , e_{p}$ be (not necessarily distinct) eigenvectors of the operator $f: V \rightarrow V$, corresponding to the eigenvalues $\lambda_{1}, \ldots , \lambda_{p}$ respectively. Prove that $e_{1} \otimes \ldots \otimes e_{p}$ is an eigenvector of the operator $f^{\otimes p}$, corresponding to the eigenvalue $\lambda_{1} \cdot \ldots \cdot \lambda_{p}$.

Let the linear operator $f: V \rightarrow V$ be reducible to diagonal form. Prove that in this case $f^{\otimes p}$ is also diagonalizable. Is the converse statement true?

Let $\lambda_{k}(k = 1, \ldots , n)$ be all the eigenvalues of the operator $f: V \rightarrow V$, taken with multiplicity, $\operatorname {dim} V = n$. Prove that the set of all eigenvalues of the operator $f^{\otimes p}$, taken with multiplicity, consists of all possible products

Prove that $\operatorname {det}\left(f^{\otimes p}\right) = (\det f)^{p n^{p-1}}$, where $f$ is an operator in $V$, $\operatorname {dim} V = n$.

Let the linear operators $f: V \rightarrow V$ and $g: W \rightarrow W$ have corresponding decompositions into invariant subspaces $V = \bigoplus_{i} V_{i}, W = \bigoplus_{j} W_{j}$. Prove that $V \otimes W = \bigoplus_{i, j} V_{i} \otimes W_{j}$ is a decomposition into invariant subspaces for $f \otimes g$.

Let $\frac{d}{d x}$ and $\frac{d}{d y}$ be the differentiation operators in the spaces of polynomials $\mathbb {R}_{n}[x]$ and $\mathbb {R}_{m}[y]$ respectively. Determine the analytic meaning of the operator $\frac{d}{d x} \otimes \frac{d}{d y}$ and find its Jordan form.

Find the Jordan form of the tensor product of two operators having the following Jordan forms: