The beginnings of tensor calculus

The Kronecker symbol. Verify that $\delta_{i}^{j}$ is an element of $\mathbb {T}_{1}^{1}(V)$, representing the identity map of the space $V$ into itself.

The metric tensor. Let $V= \left\langle \mathbf{e}_{1}, \ldots , \mathbf{e}_{n}\right\rangle_{\mathbb {R}}$ be a Euclidean vector space with inner product given by a positive definite bilinear form

(following the tradition adopted by geometers, we use here the letter $g$ and the symbols $g_{i j}$). It is customary to call $G_{0}=(g_{i j})$ the metric tensor (of type $(2,0)$) of the space $V$. Thus, the components of the metric tensor are the entries of the Gram matrix of the original basis in $V$ with respect to the corresponding inner product.

Now we recall that a Euclidean space is canonically isomorphic to its dual space $V^{*}$, and the basis $(\mathbf{e}_{i})$ is identified with its dual basis $(\mathbf{e}^{i})$ of the space $V^{*}:\left(\mathbf{e}_{i} \mid *\right)=\mathbf{e}^{i}$ (see (13) from § 1 Ch. 3). If $g^{i j}=g\left(\mathbf{e}^{i}, \mathbf{e}^{j}\right)$, then $G^{0}=\left(g^{i j}\right)$ is also called the metric tensor (now of type $(0,2)$) of the space $V$. Here $g_{i j}$ and $g^{i j}$ are the covariant and contravariant coordinates of the metric tensor $G$.

Verify that $G^{0} G_{0}=E$, i.e.,

Extension of the scalar field. Let $V$ be a real vector space with basis $(\mathbf{e}_{1}, \ldots , \mathbf{e}_{n})$, $V^{\mathbb {C}}$ its complexification (see item $3 \S 4$ Ch. 3). Since the field $\mathbb {C}$ is a vector space over $\mathbb {R}$ with basis $(1, i)$, we can construct the vector space

Verify that the $\mathbb {R}$-linear map

defines an isomorphism of $\mathbb {C} \otimes V$ with $V^{\mathbb {C}}$.

More generally: let $\mathfrak {K}$ be a subfield of a field $\mathfrak {L}, V$ a vector space over $\mathfrak {K}$. First regarding $\mathfrak {L}$ as a vector space over $\mathfrak {K}$, let us construct the tensor product $\mathfrak {L} \otimes_{\mathfrak {K}} V$. Then let us introduce on it the structure of a vector space over $\mathfrak {L}$, defining multiplication by scalars $a \in \mathfrak {L}$ by the formula

Verify the correctness of this definition.

Let $\operatorname {dim} V>1, \operatorname {dim} W>1$. Show that in the tensor product $V \otimes W$ there exist elements not representable in the form $\mathbf{v} \otimes \mathbf{w}$.

Let $A, B$ be invertible square matrices. Give the matrix inverse to $A \otimes B$.

Compare

with the block matrix (20).

Prove formula (22) for square matrices $A, B$ of orders $n$ and $m$, respectively, with complex coefficients, using the possibility of reducing them to triangular form.