Tensors

$t_{i j}=i j$;

$t_{j}^{i}=\delta_{1 i} j$;

$t^{i j}=i+j$;

$t_{i j}^{k}=2^{i+j+k^{2}}$;

$t_{k}^{i j}=\delta_{i j} \delta_{j k}$;

$t_{i j k}=\delta_{i j} \delta_{j k} \delta_{k 1}$?

$A=e^{1} \otimes e^{2}+e^{2} \otimes e^{3}+e^{2} \otimes e^{2} \in \mathbb {T}_{2}^{0}(V)$, $B=e^{1} \otimes e^{1} \otimes \left(e^{1}-e^{3}\right) \in \mathbb {T}_{3}^{0}(V)$, $v_{1}=e_{1}, \quad v_{2}=e_{1}+e_{2}, \quad v_{3}=e_{2}+e_{3}, \quad v_{4}=v_{5}=e_{2} ;$

$A=e^{1} \otimes e^{2}+e^{2} \otimes e^{3}+e^{3} \otimes e^{1} \in \mathbb {T}_{2}^{0}(V)$, $B \in \mathbb {T}_{3}^{0}(V)$, all coordinates of the tensor $B$ equal 1, $v_{1}=e_{1}+e_{2}, \quad v_{2}=e_{2}+e_{3}, \quad v_{3}=e_{3}+e_{1}, \quad v_{4}=v_{5}=e_{2}$.

$\widetilde{t}_{21}^{1}$ of the tensor $e^{1} \otimes e^{2} \otimes \left(e_{1}+e_{2}\right) \in \mathbb {T}_{2}^{1}(V)$ in the basis

$\widetilde{t}_{1}^{12}$ of the tensor $T \in \mathbb {T}_{1}^{2}(V)$, all of whose coordinates equal 1, in the basis

$\widetilde{t}_{31}^{12}$ of the tensor $e^{2} \otimes e^{1} \otimes e_{3} \otimes e_{1}+e^{3} \otimes e^{3} \otimes e_{1} \otimes e_{2} \in \mathbb {T}_{2}^{2}(V)$ in the basis

$\left(e_{1}+e_{2}\right) \otimes \left(e_{1}-e_{2}\right)$;

$\left(e_{1}+e_{2}\right) \otimes \left(e_{1}+e_{2}\right)$;

$\left(e_{1}+2 e_{2}\right) \otimes \left(e_{1}+e_{2}\right)-\left(e_{1}+e_{2}\right) \otimes \left(e_{1}+2 e_{2}\right)$;

$\left(e_{1}+2 e_{2}\right) \otimes \left(e_{3}+e_{4}\right)-\left(e_{1}-2 e_{2}\right) \otimes \left(e_{3}-e_{4}\right)$.

$f \in V^{*}$ such that $T(v, f)=0$ for every $v \in V$;

$v \in V$ such that $T(v, f)=0$ for every $f \in V^{*}$.

$\left(e^{1}+e^{2}\right) \otimes \left(e^{1}+e^{3}\right)-e^{1} \otimes e^{1}-e^{2} \otimes e^{2}$;

$\left(e^{1}-2 e^{3}\right) \otimes \left(e^{1}+3 e^{2}-e^{4}\right)+\left(e^{1}-2 e^{3}\right) \otimes e^{4}$;

$\left(e^{1}+e^{3}\right) \otimes \left(e^{2}+e^{4}\right)-\left(e^{2}-e^{4}\right) \otimes \left(e^{1}-e^{3}\right)$.

the rank of the bilinear function $u \otimes v$, where the elements $u, v \in V^{*}$ are nonzero, equals 1;

the rank of the bilinear function $\sum_{i=1}^{k} u_{i} \otimes v_{i}$, where $u_{i}, \ldots , u_{k}, v_{1}, \ldots , v_{k} \in V^{*}$, does not exceed $k$.

$\left(e_{1}+3 e_{2}-e_{3}\right) \otimes \left(e^{1}-2 e^{3}+3 e^{4}\right)-\left(e_{1}+e_{3}\right) \otimes \left(e^{1}-3 e^{3}+e^{4}\right)$;

$\left(e_{1}+2 e_{2}+3 e_{3}\right) \otimes \left(e^{1}+e^{2}-2 e^{3}\right)-\left(e_{1}-e_{2}+e_{4}\right) \otimes \left(e^{2}-2 e^{3}-3 e^{4}\right)$;

$e_{1} \otimes \left(e^{1}+e^{2}+e^{3}+e^{4}\right)+e_{2} \otimes \left(e^{1}+2 e^{2}+2 e^{3}+4 e^{4}\right)+2 e_{3} \otimes \left(e^{1}-e^{2}-e^{4}\right)$.

$t=e^{1} \otimes e_{3}, \quad v=e_{1}+e_{2}+e_{3}+e_{4}$;

$t=\left(e^{1}+e^{2}\right) \otimes \left(e_{3}+e_{4}\right), \quad v=2 e_{1}+3 e_{2}+2 e_{3}+3 e_{4}$.

$\left(2 e^{1}-e^{3}\right)$ $\otimes \left(e_{1}+e_{2}\right) ;$

$e^{1} \otimes e_{2}+\left(e^{1}+2 e^{2}\right) \otimes e_{3}$.

$e^{1} \otimes e_{3}+e^{2} \otimes e_{4}$;

$\left(e^{1}+e^{2}\right) \otimes \left(e_{3}+e_{4}\right)-\left(e^{1}+e^{3}\right) \otimes e_{3}$;

$t_{j}^{i}=\delta_{2 i}+\delta_{4 j}$;

$t_{j}^{i}=i \delta_{i j}$.

$\operatorname {tr}(\mathcal{A} \otimes \mathcal{A})$;

$\operatorname {tr}\left(\mathcal{A}^{\otimes^{k}}\right)$;

$\operatorname {det}(\mathcal{A} \otimes \mathcal{A})$.

$\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right) ;$

$\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$;

$\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$.

the intersection of the kernels $\operatorname {KerSym}$ and $\operatorname {Ker} \operatorname {Alt}$ equals zero for $q=2$ and is nonzero for $q>2$;

$\operatorname {Sym} \cdot \operatorname {Alt}=\operatorname {Alt} \cdot \operatorname {Sym}=0$;

the operator $\mathcal{P}=(\mathcal{E}-\operatorname {Sym})(\mathcal{E}-\operatorname {Alt})$ is a projection.

Find the rank of the operator $\mathcal{P}$ for $q=3$.

$S^{q}\left(V_{1} \oplus V_{2}\right) \quad$ and $\quad \oplus_{i=1}^{q} S^{i}\left(V_{1}\right) \otimes S^{q-i}\left(V_{2}\right)$;

$\Lambda^{q}\left(V_{1} \oplus V_{2}\right) \quad$ and $\quad \oplus_{i=1}^{q} \Lambda^{i}\left(V_{1}\right) \otimes \Lambda^{q-i}\left(V_{2}\right)$.

$\left(\begin{array}{lll}1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right) \quad (q=2) ; \quad$

$\left(\begin{array}{rrrr}1 & -2 & 0 & 0 \\ 1 & 4 & 0 & 0 \\ 0 & 0 & -4 & 4 \\ 0 & 0 & -3 & 1\end{array}\right) \quad (q=4)$;

$\left(\begin{array}{llll}1 & 0 & 1 & 2 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 3\end{array}\right) \quad (q=2,3)$.

$\left(\begin{array}{cccc}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1\end{array}\right) ; \quad$

$\left(\begin{array}{cccc}2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 3\end{array}\right) ;$

$\left(\begin{array}{rrrrr}2 & 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 & 0 \\ 0 & 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1\end{array}\right)$.

$\operatorname {dim} W \leqslant p$;

$\operatorname {dim} W=p$ if and only if $\xi$ is decomposable;

the smallest subspace in whose $p$-th power the $p$-vector $\xi$ lies is $U=\left\{ \xi \left(v, \ldots , v_{p-1}\right) \mid v_{i} \in V^{*}\right\}$.

$\operatorname {dim} U \geqslant p$, and $\operatorname {dim} U=p$ if and only if $\xi$ is decomposable.