Math Problems Vault

Problem 35.1

(1)(2)(3)

Let $\xi^{1}, \xi^{2}$ and $\eta^{1}, \eta^{2}$ be the coordinates of vectors $x$ and $y$ in an arbitrary basis of a two-dimensional vector space. Assign to this basis the numbers:

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(1)

$\xi^{1} + \xi^{2}$

(2)

$\xi^{1} + \eta^{1}$

(3)

$\left[\begin{smallmatrix} \xi^{1} & \eta^{1} \\ \xi^{2} & \eta^{2}\end{smallmatrix}\right]$

How do these numbers change under a change of basis? Check whether each of the given quantities is a tensor.

Problem 35.2

(1)(2)

Assign to each basis in the vector space $\mathcal{L}_{n}$ :

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(1)

the number $1$

(2)

the ordered set of numbers $1, \ldots , n$ . Will this correspondence be a tensor? An invariant?

Problem 35.3

Let $\varphi$ be a linear transformation of a vector space $\mathcal{L}_{3}$ . Denote by $A = \left\| a_{j}^{i}\right\|$ its matrix in an arbitrary basis. In which cases do the indicated quantities define an invariant:

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(1)

$\operatorname {det} A$

(2)

$\cos \operatorname {det} A$

(3)

$\operatorname {Rg} A$

(4)

$\operatorname {det} A^{T} A$

(5)

$a_{1}^{1} + a_{2}^{2}$

(6)

$a_{1}^{1} + a_{2}^{2} + a_{3}^{3}$

Problem 35.4

Let b be a bilinear function, and let $B = \left\| b_{ij}\right\|$ be its matrix in an arbitrary basis of the space $\mathcal{L}_{n}$ . How does each of the indicated quantities change under a change of basis? In which cases does it define an invariant?

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(1)

$\operatorname {det} B$

(2)

$b_{11} + \ldots + b_{nn}$

(3)

$b_{11}$

(4)

$\operatorname {det} B^{T} B$

(5)

$\operatorname {Rg} B$

(6)

$\operatorname {sign} \operatorname {det} B$ .

Problem 35.5

(1)(2)

Let $\mathrm{f}$ be a linear function on a vector space $\mathcal{L}_{n}$ and let $\left(a_{1}, \ldots , a_{n}\right)$ be the row of its coefficients in an arbitrary basis. Assign to this basis:

?

(1)

the number $a_{1} + \ldots + a_{n}$

(2)

the ordered set of numbers $a_{1}, \ldots , a_{n}$ .

How do these quantities change under a change of basis? Which of them are tensors? Invariants?

Problem 35.6

(1)(2)

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(1)

Of what type is the tensor in $\mathcal{L}_{n}$ that a bilinear function defines? How can the components of this tensor be found?

(2)

Of what type is the tensor in $\mathcal{L}_{n}$ that a quadratic function defines? How can the components of this tensor be found?

Problem 35.7

(1)(2)

Linear functions $f, g$ have, in a basis $e$ of the space $\mathcal{L}_{n}$ , coefficients $a_{1}, \ldots , a_{n}$ and $b_{1}, \ldots , b_{n}$ respectively. Show that the functions:

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(1)

$\mathrm{f}^{2}$

(2)

$\mathrm{fg}$

define tensors in $\mathcal{L}_{n}$ ; indicate their types and write out the components of each in the basis e.

Problem 35.8

(1)(2)

Linear functions $\mathrm{f}, \mathrm{g}$ have, in a basis $e$ of the space $\mathcal{L}_{n}$ , coefficients $a_{1}, \ldots , a_{n}$ and $b_{1}, \ldots , b_{n}$ respectively. Assign to each pair of vectors $x, y$ from $\mathcal{L}_{n}$ the number:

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(1)

$\mathrm{f}(x) \mathrm{g}(y)$

(2)

$\mathrm{f}(x) \mathrm{f}(y)$ .

Show that each of the resulting functions defines a tensor in $\mathcal{L}_{n}$ ; indicate its type and write out the components in the basis $\mathbf{e}$ .

Problem 35.9

(1)(2)

To each pair of vectors $x, y$ of a vector space $\mathcal{L}_{n}(n \geqslant 3)$ a number $\mathrm{f}(x, y)$ is assigned, defined in terms of the components $\xi^{1}, \ldots , \xi^{n}$ and $\eta^{1}, \ldots , \eta^{n}$ of these vectors, given in a basis $\mathbf{e}$ , by one of the following formulas:

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(1)

$\mathrm{f}(x, y) = \xi^{1} \eta^{3}$

(2)

$\mathrm{f}(x, y) = \sum_{1}^{n} \xi^{i} \eta^{i}$ .

Indicate the type of the corresponding tensor and write out its components in the basis $\mathbf{e}$ .

Problem 35.10

(1)(2)(3)(4)

A function f: $\mathcal{L}_{n} \rightarrow \mathbb {R}(n \geqslant 2)$ is defined in terms of the components $\xi^{1}, \ldots , \xi^{n}$ of a vector $x$ , given in a basis $\mathbf{e}$ , by one of the formulas:

?

(1)

$f(x) = \xi^{1} + \xi^{2}$

(2)

$f(x) = \left(\xi^{1}\right)^{2} + 2\xi^{1} \xi^{2}$

(3)

$f(x) = \left(\xi^{1} + \ldots + \xi^{n}\right)^{2}$

(4)

$f(x) = \sum_{1}^{n}\left(\xi^{i}\right)^{2}$ .

Indicate the type of the corresponding tensor and write out its components in the basis $\mathbf{e}$ .

Problem 35.11

Let $\mathcal{L}_{n}^{ * }$ be the space of all linear functions defined on a vector space $\mathcal{L}_{n}$ , and let $\varphi : \mathcal{L}_{n}^{ * } \rightarrow \mathbb {R}$ be a linear function on $\mathcal{L}_{n}^{ * }$ . Show that $\varphi$ defines a tensor of type $(1, 0)$ on $\mathcal{L}_{n}$ .

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Problem 35.12

(1)(2)(3)(4)

Given tensors $a_{ij}, a_{j}^{i}, \xi^{i}, \eta^{i}, b_{i}$ . The quantities $c, d, g, h$ are defined in each basis by the formulas:

?

(1)

$c = a_{ij} \xi^{i} \eta^{j}$

(2)

$d = a_{ij} \xi^{i} \xi^{j}$

(3)

$g = a_{j}^{i} b_{i} \xi^{j}$

(4)

$h = b_{i} \xi^{i}$ .

Relying on the transformation law of the components of the given tensors, show directly that these quantities are invariants.

Problem 35.13

Given tensors $a_{j}^{i}, \xi^{i}, b_{i}$ . The quantities $c^{i}, d_{i}$ are defined in each basis by the formulas $c^{i} = a_{j}^{i} \xi^{j}$ and $d_{i} = a_{i}^{j} b_{j}$ respectively. Relying on the transformation law of the components of the given tensors, show that $c^{i}$ is a vector, and $d_{i}$ is a covector.

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Problem 35.14

A tensor of type $(1, 1)$ has, in some basis, components

\delta _{j}^{i} = \begin{cases} 1, & \text{ if } i = j \\ 0, & \text{ if } i \neq j \end{cases}

Do its components change on passing to another basis? What is the geometric meaning of this tensor?

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Problem 35.15

A tensor of type $(0, 2)$ has, in some basis, components

\delta _{ij} = \begin{cases} 1, & \text{ if } i = j \\ 0, & \text{ if } i \neq j \end{cases}

How will its components change on passing to another basis? What bilinear function corresponds to this tensor?

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Problem 35.16

A tensor of type $(1, 0)$ has, in some basis, components

\theta ^{i} = \begin{cases} 1, & \text{ if } i = i_{0} \\ 0, & \text{ if } i \neq i_{0} \end{cases}

( $i_{0}$ is a fixed integer, $1 \leqslant i_{0} \leqslant n$ ). Find the components of this tensor in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ .

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Problem 35.17

A tensor of type $(0, 1)$ has, in some basis, components

\theta _{i} = \begin{cases} 1, & \text{ if } i = i_{0} \\ 0, & \text{ if } i \neq i_{0} \end{cases}

( $i_{0}$ is a fixed integer, $1 \leqslant i_{0} \leqslant n$ ). Find the components of this tensor in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ .

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Problem 35.18

To each basis of the space $\mathcal{L}_{n}(n > 2)$ are assigned the numbers

\delta _{kl}^{ij} = \begin{cases} 1, & \text{ if } i = k \neq j = l \\ - 1, & \text{ if } i = l \neq j = k; \\ 0 & \text{ in the remaining cases. } \end{cases}

Will such a correspondence be a tensor? How many zero components does this tensor have for $n = 3$ ?

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Problem 35.19

(1)(2)

A tensor $\theta$ of type $(0, 2)$ has, in some basis $e$ of a vector space $\mathcal{L}_{n}(n > 2)$ , components $\theta_{kl} = \delta_{kl}^{i_{0} j_{0}}$ ( $i_{0}, j_{0}$ are fixed integers, $1 \leqslant i_{0} \leqslant n, 1 \leqslant j_{0} \leqslant n$ , the symbol $\delta_{kl}^{i_{0} j_{0}}$ is defined in problem 35. 18).

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(1)

Write out explicitly all the components of the tensor $\theta$ in the basis $e$ for $n = 3$ .

(2)

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

Problem 35.20

To each basis of the space $\mathcal{L}_{n}$ $(n > k \geqslant 1)$ are assigned the numbers:

\delta _{j_1 \ldots j_k}^{i_1 \ldots i_k} = \begin{cases} 1, & \text{if } (i_1, \ldots , i_k) \text{ is an even permutation of the pairwise distinct numbers } j_1, \ldots , j_k; \\ -1, & \text{if } (i_1, \ldots , i_k) \text{ is an odd permutation of the pairwise distinct numbers } j_1, \ldots , j_k; \\ 0, & \text{in the remaining cases}. \end{cases}

Will such a correspondence be a tensor?

?

Problem 35.21

(1)(2)

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(1)

A tensor of type $(0, n)$ has, in some basis, components $\varepsilon_{i_{1} \ldots i_{n}} = \begin{cases} (- 1)^{N\left(i_{1} \ldots i_{n}\right)}, & \text{ if all the numbers } i_{1}, \ldots , i_{k} \text{ are distinct; } \\ 0 & \text{ in the remaining cases } \end{cases}$ ( $N(i_{1} \ldots i_{n})$ is the number of inversions in the permutation $\left(i_{1}, \ldots , i_{n}\right)$ ). Compute the components of this tensor in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ .

(2)

To each basis of the space $\mathcal{L}_{n}$ are assigned the numbers $\varepsilon_{i_{1}}, \ldots , i_{n}$ . Will this correspondence be a tensor of type $(0, n)$ ?

Problem 35.22

In a four-dimensional space a trivalent tensor is given. How many components does it have? How many terms enter into the expression of a new component in terms of the old ones when writing the transformation law of the components? How many factors will there be in each term?

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Problem 35.23

(1)(2)(3)

In the space $\mathcal{L}_{2}$ a tensor of type is given:

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(1)

$(1, 1)$

(2)

$(2, 0)$

(3)

$(1, 2)$

In expanded form, without using the shortened summation notation, write the transformation law of its components.

Problem 35.24

(1)(2)(3)

In a two-dimensional space a tensor of type $(p, q)$ is given. Order its components so that they form a column a of height $2^{p + q}$ . Write the transformation law of the tensor components in the form $\mathbf{a}^{\prime } = V\mathbf{a}$ , where $V$ is a square matrix of order $2^{p + q}$ and:

?

(1)

$p = 1, q = 1$

(2)

$p = 2, q = 0$

(3)

$p = 1, q = 2$ .

Problem 35.25

(1)(2)(3)

Write in matrix form the transformation law of the components of tensors of type:

?

(1)

$(0, 2)$

(2)

$(1, 1)$

(3)

$(2, 0)$

Problem 35.26

(1)(2)(3)

The components of a bivalent tensor of type $(p, q)$ form, in an arbitrary basis $e$ of a vector space $\mathcal{L}_{n}$ , a matrix $A_{e}$ . Assign to the basis $e$ the matrix $A_{e}^{ - 1}$ . Prove that this correspondence defines a tensor, and indicate its type, if:

?

(1)

$p = 0, q = 2$

(2)

$p = 1, q = 1$ (explain the geometric meaning of the resulting tensor)

(3)

$p = 2, q = 0$ .

Problem 35.27

Tensors of which types have two-dimensional matrices of components? Three-dimensional? Four-dimensional? $k$ -dimensional matrices of components?

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Problem 35.28

A three-dimensional matrix $\left\| a_{ijk}\right\|$ of order two has the section $k = 1$ consisting of ones, and the section $k = 2$ of zeros. Write out $a_{ijk}$ for all possible values of the indices.

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Problem 35.29

A three-dimensional matrix $\left\| a_{ijk}\right\|$ of order three has the sections $k = 1$ and $k = 2$ consisting of ones, and the section $k = 3$ of zeros. Write out the two-dimensional sections of this matrix corresponding to $i = 1, i = 2, i = 3$ .

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Problem 35.30

(1)(2)(3)

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(1)

How many distinct two-dimensional sections does a three-dimensional matrix of order three have? What order does each section have?

(2)

How many two-dimensional sections does a four-dimensional matrix of order two have?

(3)

How many two-dimensional sections does a four-dimensional matrix of order three have?

Problem 35.31

(1)(2)

The numbers $\delta_{kl}^{ij}$ form a four-dimensional matrix of order two.

?

(1)

Write out all its two-dimensional sections corresponding to fixed lower indices.

(2)

Find the relation between the sections of the matrix $\left\| \delta_{kl}^{ij}\right\|$ and the matrix $\left\| \theta_{kl}\right\|$ (the symbols $\delta_{kl}^{ij}, \theta_{kl}$ are defined in problems $35. 18, 35. 19$ respectively).

Problem 35.32

(1)(2)

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(1)

Given a basis $e$ and a $(p + q)$ -dimensional matrix $A$ . Prove that there exists a tensor of type $(p, q)$ having matrix $A$ in the basis $\mathbf{e}$ .

(2)

Prove that a tensor of any prescribed type exists.

Problem 35.33

(1)(2)(3)

Let $\mathrm{f}$ be a real function of three arguments $x\in \mathcal{L}_{n}, y\in \mathcal{L}_{n}, z\in \mathcal{L}_{n}$ , linear in each of these arguments when the others are fixed.

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(1)

Express the value of this function in terms of the components of the vectors $x, y, z$ .

(2)

Show that the collection of coefficients of the resulting form constitutes a tensor of type $(0, 3)$ .

(3)

Express the components of this tensor in terms of the values of $f$ on the basis vectors.

Problem 35.34

(1)(2)(3)

Linear functions $\mathrm{f}, \mathrm{g}, \mathrm{h}$ on $\mathcal{L}_{3}$ have, in a basis $\mathbf{e}$ , coefficients $\alpha_{1}, \alpha_{2}, \alpha_{3}; \beta_{1}, \beta_{2}, \beta_{3}; \gamma_{1}, \gamma_{2}, \gamma_{3}$ respectively. Assign to a triple of vectors $x, y, z$ from $\mathcal{L}_{3}$ the number:

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(1)

$\mathrm{f}(x) \mathrm{g}(y) \mathrm{h}(z)$

(2)

$\mathrm{f}(x) \mathrm{f}(y) \mathrm{f}(z)$

(3)

$\mathrm{f}(x) \mathrm{f}(y) \mathrm{f}(z) + \mathrm{g}(x) \mathrm{g}(y) \mathrm{g}(z) + \mathrm{h}(x) \mathrm{h}(y) \mathrm{h}(z)$ .

Show that each of the constructed functions defines a tensor in $\mathcal{L}_{3}$ ; indicate its type and write out the matrix in the basis $\mathbf{e}$ .

Problem 35.35

(1)(2)

To each triple of vectors of the space $\mathcal{L}_{3}$ a number $\mathrm{f}(x, y, z)$ is assigned, defined in terms of the components of these vectors in some basis — $\xi^{1}, \xi^{2}, \xi^{3}; \eta^{2}, \eta^{2}, \eta^{3}; \zeta^{1}, \zeta^{2}, \zeta^{3}$ — by one of the following formulas:

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(1)

$\mathrm{f}(x, y, z) = \xi^{1} \eta^{2} \zeta^{3} + \xi^{3} \eta^{2} \zeta^{1}$

(2)

$\mathrm{f}(x, y, z) = \sum_{i = 1}^{3} \xi^{i} \eta^{i} \zeta^{i}$

Indicate the type of the corresponding tensor and write out its matrix.

Problem 35.36

Let $\mathcal{B}$ be the vector space of bilinear functions on $\mathcal{L}_{n}$ , and let $\varphi : \mathcal{L}_{n} \rightarrow \mathcal{B}$ be a linear mapping. Show that $\varphi$ defines a tensor of type $(0, 3)$ in the space $\mathcal{L}_{n}$ .

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Problem 35.37

(1)(2)(3)

A tensor of type $(p, q)$ in the basis $e_{1}, e_{2}, e_{3}$ of the space $\mathcal{L}_{3}$ is given by the matrix $A$ . Find its matrix in the basis $e_{1}^{\prime }, e_{2}^{\prime }, e_{3}^{\prime }$ , if:

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(1)

$p = 2, q = 1, A = \mathbf{A}_{726}, e_{1}^{\prime } = e_{1}, e_{2}^{\prime } = e_{3}, e_{3}^{\prime } = e_{2}$ . $\mathbf{A}_{726} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 & 3 \\ 4 & 3 & 2 \\ 9 & 8 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 5 & 6 \\ 1 & 5 & 9 \\ 6 & 5 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 7 & 8 & 9 \\ 8 & 7 & 6 \\ 3 & 2 & 1 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(2)

$p = 2, \quad q = 1, \quad A = \mathbf{A}_{727}, \quad e_{1}^{\prime } = - e_{1}, \quad e_{2}^{\prime } = - e_{2}, \quad e_{3}^{\prime } = - e_{3}$ . $\mathbf{A}_{727} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 3 & 2 \\ 9 & 7 & 8 \\ 4 & 2 & 3 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 7 & 9 & 8 \\ 3 & 1 & 2 \\ 8 & 6 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 5 \\ 6 & 4 & 5 \\ 1 & 9 & 5 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(3)

$p = 0, \quad q = 3, \quad A = \mathbf{A}_{723}, \quad e_{1}^{\prime } = 2e_{1}, \quad e_{2}^{\prime } = - e_{2}, \quad e_{3}^{\prime } = 3e_{3}$ . $\mathbf{A}_{723} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & -2 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 & -2 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Problem 35.38

Determine how the components of a tensor of type $(1, 2)$ , given in the space $\mathcal{L}_{n}$ , change under an arbitrary permutation of the basis vectors.

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Problem 35.39

(1)(2)(3)(4)

A tensor of type $(p, q)$ in the basis $e$ of the space $\mathcal{L}_{2}$ is given by the matrix $A$ . Find its matrix in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ , if:

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(1)

$p = 1, q = 2, A = \mathbf{A}_{652}, S = \mathbf{A}_{24}$ . $\mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$ .

(2)

$p = 0, q = 3, A = \mathbf{A}_{654}, S = \mathbf{A}_{19}$ . $\mathbf{A}_{654} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & -1 \\ 2 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

(3)

$p = 0, q = 3, A = \mathbf{A}_{654}, S = \mathbf{A}_{24}$ . $\mathbf{A}_{654} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & -1 \\ 2 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$ .

(4)

$p = 2, q = 1, A = \mathbf{A}_{654}, S = \mathbf{A}_{24}$ . $\mathbf{A}_{654} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & -1 \\ 2 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$ .