Linear operations, tensor multiplication (36.1-36.20)

$A = \mathbf{A}_{650}, B = \mathbf{A}_{651}$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$A = \mathbf{A}_{651}, B = \mathbf{A}_{652}$. $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$A = \mathbf{A}_{652}, B = \mathbf{A}_{653}$. $\mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{653} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 7 & -4 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 8 \\ 2 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$A = \mathbf{A}_{693}, B = \mathbf{A}_{697}$. $\mathbf{A}_{693} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -1 & -2 \\ -3 & -4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right], \mathbf{A}_{697} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} 1 & 1 \\ 2 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 2 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$A = \mathbf{A}_{662}, B = \mathbf{A}_{665}, C = \mathbf{A}_{663}, D = \mathbf{A}_{652}$. $\mathbf{A}_{662} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 11 & 14 \\ 8 & 13 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 17 & 23 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{665} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -6 & 4 \\ 13 & 22 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 23 \\ 17 & 27 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{663} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 0 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 5 \\ -1 & 5 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$A = \mathbf{A}_{650}, B = \mathbf{A}_{651}, C = \mathbf{A}_{652}, D = \mathbf{A}_{653}$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{653} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 7 & -4 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 8 \\ 2 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$A = \mathbf{A}_{666}, B = \mathbf{A}_{663}, C = \mathbf{A}_{650}, D = \mathbf{A}_{652}$. $\mathbf{A}_{666} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -10 & -6 \\ -1 & -8 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -15 & -11 \\ 3 & 1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{663} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 0 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 5 \\ -1 & 5 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

What is the dimension of the vector space $\mathcal{L}$ of tensors of type $(p, q)$ in a two-dimensional space?

Indicate some basis in $\mathcal{L}$.

Indicate another basis in $\mathcal{L}$.

$(0, 1)$

$(1, 1)$

(р) $(0, 2)$

$(1, 2)$

Type of $a$: $(1, 0)$, matrix of $a$: $\mathbf{c}_{12}$, type of $b$: $(1, 0)$, matrix of $b$: $\mathbf{c}_{31}$. $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$.

Type of $a$: $(1, 0)$, matrix of $a$: $\mathbf{c}_{12}$, type of $b$: $(0, 1)$, matrix of $b$: $\mathbf{c}_{31}^T$. $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$.

Type of $a$: $(1, 0)$, matrix of $a$: $\mathbf{c}_{12}^T$, type of $b$: $(1, 0)$, matrix of $b$: $\mathbf{c}_{31}$. $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$.

Type of $a$: $(0, 1)$, matrix of $a$: $\mathbf{c}_{12}^T$, type of $b$: $(0, 1)$, matrix of $b$: $\mathbf{c}_{31}^T$. $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$.

Type of $a$: $(0, 2)$, matrix of $a$: $\mathbf{A}_{17}$, type of $b$: $(0, 1)$, matrix of $b$: $\mathbf{c}_{7}^T$. $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right]$.

Type of $a$: $(0, 1)$, matrix of $a$: $\mathbf{c}_{7}^T$, type of $b$: $(0, 2)$, matrix of $b$: $\mathbf{A}_{17}$. $\mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$.

Type of $a$: $(2, 0)$, matrix of $a$: $\mathbf{A}_{18}$, type of $b$: $(1, 0)$, matrix of $b$: $\mathbf{c}_{8}$. $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$.

Type of $a$: $(1, 1)$, matrix of $a$: $\mathbf{A}_{18}$, type of $b$: $(1, 0)$, matrix of $b$: $\mathbf{c}_{8}$. $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$.

Type of $a$: $(1, 0)$, matrix of $a$: $\mathbf{c}_{8}$, type of $b$: $(2, 0)$, matrix of $b$: $\mathbf{A}_{18}$. $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$.

Type of $a$: $(1, 0)$, matrix of $a$: $\mathbf{c}_{8}$, type of $b$: $(1, 1)$, matrix of $b$: $\mathbf{A}_{18}$. $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$.

Type of $a$: $(0, 3)$, matrix of $a$: $\mathbf{A}_{650}$, type of $b$: $(0, 1)$, matrix of $b$: $\mathbf{c}_{25}^T$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{c}_{25} = \left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right]$.

Type of $a$: $(0, 1)$, matrix of $a$: $\mathbf{c}_{25}^T$, type of $b$: $(0, 3)$, matrix of $b$: $\mathbf{A}_{650}$. $\mathbf{c}_{25} = \left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right], \mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

Type of $a$: $(1, 2)$, matrix of $a$: $\mathbf{A}_{651}$, type of $b$: $(0, 1)$, matrix of $b$: $\mathbf{c}_{8}^T$. $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$.

Type of $a$: $(0, 1)$, matrix of $a$: $\mathbf{c}_{8}^T$, type of $b$: $(1, 2)$, matrix of $b$: $\mathbf{A}_{651}$. $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

Type of $a$: $(0, 2)$, matrix of $a$: $\mathbf{A}_{17}$, type of $b$: $(0, 2)$, matrix of $b$: $\mathbf{A}_{18}$. $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$.

Type of $a$: $(0, 2)$, matrix of $a$: $\mathbf{A}_{18}$, type of $b$: $(0, 2)$, matrix of $b$: $\mathbf{A}_{17}$. $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$.

Type of $a$: $(1, 1)$, matrix of $a$: $\mathbf{A}_{13}$, type of $b$: $(1, 1)$, matrix of $b$: $\mathbf{A}_{19}$. $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$.

Type of $a$: $(2, 0)$, matrix of $a$: $\mathbf{A}_{13}$, type of $b$: $(1, 1)$, matrix of $b$: $\mathbf{A}_{19}$. $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$.

$a^{i} b^{j}$

$a_{i} b_{j}$

$a^{i} b_{j}$

$a_{i} b^{j}$

$A\otimes B$

$B\otimes A?$

$f\otimes g$;

$g\otimes f$.

Let $x$ be a vector, and $\mathrm{f}$ a covector. Prove that $\mathrm{f} \otimes x = x\otimes \mathrm{f}$.

Give an example of tensors $a$ and $b$ for which $a\otimes b \neq b\otimes a$.

$x_{1} \otimes x_{2} + x_{2} \otimes x_{3}$

$x_{1} \otimes x_{2} \otimes x_{3} + x_{2} \otimes x_{3}$

$x_{1} \otimes \mathrm{f}_{1} - 2\mathrm{f}_{1} \otimes x_{1}$

$x_{1} \otimes \mathrm{f}_{2} + \mathrm{f}_{1} \otimes \mathrm{f}_{2}$

$x_{1} \otimes \mathrm{f}_{2} + x_{2} \otimes \mathrm{f}_{1}$

$f_{1} \otimes x_{1} \otimes x_{2} + x_{2} \otimes x_{3} \otimes \mathrm{f}_{2}$

$x_{1} \otimes x_{2} + x_{3} \otimes x_{3} - x_{1} \otimes x_{1}$

$f_{1} \otimes f_{2} - 3\left(f_{2} \otimes f_{3}\right)$.

Let $a = x\otimes y$, and let the vectors $x$ and $y$ have, in the basis $\mathbf{e}$, components $1, 0, 0$ and $0, 1, 0$ respectively. Find the components of the tensor $a$ in the basis $e$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$, where $S = \mathbf{A}_{207}$. $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$.

Let $a = \mathrm{f} \otimes \mathrm{g}$, and let the covectors $\mathrm{f}$ and $\mathrm{g}$ have, in the basis $e$, coordinate rows $(1, 0, 0)$ and $(0, 1, 0)$ respectively. Find the components of the tensor $a$ in the basis $\mathbf{e}$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$, where $S = \mathbf{A}_{207}$. $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$.

Let $a = x\otimes \mathrm{f}$, and let the vector $x$ and covector $\mathrm{f}$ have, in the basis, components $1, 0, 0$ and $0, 1, 0$ respectively. Find the components of the tensor $a$ in the basis $e$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$, where $S = \mathbf{A}_{207}$. $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$.

type $(2, 0)$ and matrix $\mathbf{A}_{5}$. $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$.

type $(2, 1)$ and matrix $\mathbf{A}_{673}$. $\mathbf{A}_{673} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & -1 \\ 2 & -2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & -1 \\ 2 & -2 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

Let $a$ be a tensor of type $(1, 1)$ whose component matrix has rank $r$. Prove that there exist $r$ linearly independent vectors $a_{1}, \ldots , a_{r}$ and $r$ linearly independent covectors $\mathrm{f}_{1}, \ldots , \mathrm{f}_{r}$ such that $a = \sum_{\alpha = 1}^{r} a_{\alpha } \otimes \mathrm{f}^{\alpha }$.

State and prove an analogous statement for tensors of type $(2, 0)$.

Let a tensor $a$ of type $(0, 2)$ have, in some basis, a matrix of rank $r$. Prove that there exist $r$ linearly independent covectors $\mathrm{f}_{1}, \ldots , \mathrm{f}_{r}$ and $r$ linearly independent covectors $\mathrm{g}_{1}, \ldots , \mathrm{g}_{r}$ such that $a = \sum_{\alpha = 1}^{r} \mathrm{f}_{\alpha } \otimes \mathrm{g}_{\alpha }$.

Represent the bilinear function $3\xi^{1} \eta^{1} + 2\xi^{1} \eta^{2} + 3\xi^{2} \eta^{1} + 2\xi^{2} \eta^{2}$ as a product of linear ones. Is such a representation unique?

A bilinear function $\mathrm{f}$ is given, in some basis of a vector space, by the matrix $\mathbf{A}_{454}$. Represent it as a sum of two products of pairs of linear functions: $\mathrm{f}(x, y) = \mathrm{f}_{1}(x) \mathrm{g}_{1}(y) + \mathrm{f}_{2}(x) \mathrm{g}_{2}(y)$. Is this representation unique? $\mathbf{A}_{454} = \left[\begin{smallmatrix} 2 & 4 & 6 & 1 \\ 1 & 2 & 3 & 1 \\ -3 & -6 & -9 & 1 \\ 1 & 2 & 3 & 1 \end{smallmatrix}\right]$.