§ 1

Let $\mathcal{B}$ be the linear space of bilinear functions defined on an $n$-dimensional linear space $\mathcal{L}$, and let $\mathrm{A}$ be a linear transformation of the space $\mathcal{B}$. Prove that $\mathrm{A}$ is a tensor of type $(2,2)$ in the space $\mathcal{L}$.

A tensor of type $(0, n)$ in an $n$-dimensional linear space has in a basis $\mathbf{e}$ the components $\varepsilon_{i_{1} \ldots i_{n}}=0$, if among the values $i_{1}, \ldots , i_{n}$ there are equal ones, and

otherwise. Find the components of this tensor in the basis $\mathbf{e}^{\prime }=\mathbf{e} S$.

A linear function $\mathrm{f}$ is given in a basis $\mathbf{e}$ by a row $\mathbf{f}$, and a vector $a$ by a column $\mathbf{a}$. Find the matrix of the tensor $a \otimes \mathrm{f}$. What geometric meaning does this tensor have?

How many different tensors can be formed by means of contraction from a tensor of type $(2,2)$ ?

Prove that the tensor from problem 3 is antisymmetric in any subset of the set of indices.

Prove that for any tensor of type $(1,1)$ the equalities

hold.