Basic Concepts

A tensor of type $(1,0)$ has components in the basis $e_{1}, \ldots , e_{n}$ ($i_{0}$ is a fixed number, $1 \leqslant i_{0} \leqslant n$)

Find the components of this tensor in the new basis $e_{1}^{\prime }, \ldots , e_{n}^{\prime }$, given the transition matrix $C = \left(c_{j}^{i}\right)$ from the old basis to the new one.

A tensor of type $(0,1)$ has components in the basis $e_{1}, \ldots , e_{n}$ ($i_{0}$ is a fixed number, $1 \leqslant i_{0} \leqslant n$)

Find the components of this tensor in the new basis $e_{1}^{\prime }, \ldots , e_{n}^{\prime }$, given the transition matrix $C = \left(c_{j}^{i}\right)$ from the old basis to the new one.

Verify that a vector (respectively a covector (linear function), a linear operator, a bilinear form) on $V$ defines a tensor of type $(1,0)$ (respectively $(0,1),(1,1),(0,2)$).

Verify that the maps defined above between the spaces of tensors $\mathbf{T}_{q}^{p}$ and multilinear functions $\mathbf{P}_{q}^{p}$ are well-defined, linear, and bijective. How is the operation $\otimes$ written in terms of $\mathbf{P}_{q}^{p}$?

Verify that the tensor product and contraction are well-defined tensor operations.

Show that assigning the Kronecker symbol $\delta_{j}^{i}$ to every basis defines a tensor of type $(1,1)$. Which linear operator and which multilinear function correspond to it?

Show that $\varepsilon_{i_{1} \ldots i_{n}}$ is not a tensor.

Let $v \in V, \varphi \in V^{*}$. Find the image of the operator corresponding to the tensor $v \otimes \varphi$ in the space $V$.

Fix a basis $e_{1}, e_{2}, e_{3}$ in $V$ and define the tensor $A = e^{1} \otimes e_{2}-e^{2} \otimes e_{3}+e^{3} \otimes e_{1} \in \mathbf{T}_{1}^{1}$ and the trilinear map $F \in \mathbf{P}_{3}^{0}$, $F\left(v_{1}, v_{2}, v_{3}\right) = \det \left\| v_{j}^{i}\right\|$, where $v_{j}^{i}$ is the $i$-th coordinate of the vector $v_{j}$ in the given basis. Compute $\left(A \otimes T_{F}\right)_{2133}^{1}$ and $\left(T_{F} \otimes A\right)_{2133}^{1}$.

Let, in the basis $e_{1}, e_{2}, e_{3}$, the components of the tensor $T \in \mathbf{T}_{3}^{2}(V)$ be defined as $T_{k s m}^{i j} = 2 i$. Find $\left(T^{\prime }\right)_{223}^{12}$, if $e_{i}^{\prime } = \sum_{j \leqslant i}(i-j+1) e_{j}$.

Show that a tensor of type $(1,1)$, invariant under all orthogonal changes of coordinates in $\mathbb {R}^{n}$, is proportional to the tensor $\delta_{j}^{i}$.

Show that a tensor of the third valence, invariant under arbitrary changes of coordinates, is equal to zero.

Find the general form of a tensor of the fourth valence, invariant under an arbitrary change of coordinates. Consider the various cases $(p, q), p+q = 4$.

Express the trace of the operator A as the result of tensor operations.

Express the determinant of the matrix of an operator as the result of tensor operations.

Let $A$ be a linear operator. Find expressions for the quantities $A_{i}^{i}, A_{j}^{i} A_{i}^{j}, A_{j}^{i} A_{k}^{j} A_{i}^{k}$ in terms of the roots and coefficients of the characteristic polynomial $\operatorname {det}\left(A-\lambda E\right)$.

Consider the multilinear function $F \in \mathbf{P}_{n}^{n}(V), \operatorname {dim} V = n$, given by the formula

For a given basis $e_{1}, \ldots , e_{n}$, find the components of $T_{F}$.

Let $\varphi \in \mathbf{P}_{q}^{p}$. Show that the value of $\varphi$ on a specific set of covector arguments $\xi^{j}$ and vector arguments $v_{i}$, i.e. $\varphi \left(\xi^{1}, \ldots , \xi^{p}, v_{1}, \ldots , v_{q}\right)$, coincides with the $(p+q)$-fold contraction of the tensor $T_{\varphi } \otimes \xi^{1} \otimes \ldots \otimes \xi^{p} \otimes v_{1} \otimes \ldots \otimes v_{q}$ of type $(p+q, p+q)$.

Let $\mathbb {R}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with real coefficients. Verify that the map $l: \mathbb {R}_{n}[x] \rightarrow \mathbb {R}$, given by each of the following methods: