Math Problems Vault

Problem 38.1

A function $\mathrm{f}_{a}$ of two vectors on a three-dimensional vector space assigns to any vectors $x$ and $y$ the scalar triple product $(a, x, y)$ . Prove that $\mathrm{f}_{a}$ is a $2$ -form. Express its matrix in an orthonormal basis in terms of the coordinates of the vector $a$ .

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Problem 38.2

Find the relation between the cross product of two vectors and their exterior product.

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Problem 38.3

(1)(2)(3)

Write the matrix of the $2$ -form $\omega$ in the basis $e$ of the space $\mathcal{L}_{4}$ , if its expression is given in terms of $1$ -forms forming the basis biorthogonal to e:

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(1)

$\omega = \mathrm{f}^{1} \wedge \mathrm{f}^{2}$

(2)

$\omega = - \mathrm{f}_{1} \wedge \mathrm{f}^{3} + \mathrm{f}^{2} \wedge \mathrm{f}^{4}$

(3)

$\omega = \mathrm{f}^{1} \wedge \mathrm{f}^{2} + \mathrm{f}^{1} \wedge \mathrm{f}^{3} + \mathrm{f}^{1} \wedge \mathrm{f}^{4} + \mathrm{f}^{2} \wedge \mathrm{f}^{3} + \mathrm{f}^{3} \wedge \mathrm{f}^{4}$

Problem 38.4

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

Find the exterior product of two $1$ -forms given by coordinate rows:

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(1)

$\mathbf{c}_{79}^{T}, \mathbf{c}_{75}^{T}$ . $\mathbf{c}_{79} = \left[\begin{smallmatrix} 20 \\ -2 \\ 42 \end{smallmatrix}\right], \mathbf{c}_{75} = \left[\begin{smallmatrix} 1 \\ 2 \\ 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{c}_{95}^{T}, \mathbf{c}_{93}^{T}$ . $\mathbf{c}_{95} = \left[\begin{smallmatrix} -2 \\ -2 \\ -4 \end{smallmatrix}\right], \mathbf{c}_{93} = \left[\begin{smallmatrix} 1 \\ 2 \\ 2 \end{smallmatrix}\right]$ .

(3)

$\mathbf{c}_{174}^{T}, \mathbf{c}_{166}^{T}$ . $\mathbf{c}_{174} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{156}^{T}, \mathbf{c}_{193}^{T}$ . $\mathbf{c}_{156} = \left[\begin{smallmatrix} 6 \\ 6 \\ 0 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{193} = \left[\begin{smallmatrix} 0 \\ 0 \\ 0 \\ 1 \end{smallmatrix}\right]$ .

Problem 38.5

Find the exterior product of three $1$ -forms given by coordinate rows:

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(1)

$\mathbf{c}_{12}^{T}, \mathbf{c}_{13}^{T}, \mathbf{c}_{14}^{T}$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{13} = \left[\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{14} = \left[\begin{smallmatrix} 7 \\ -1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{c}_{99}^{T}, \mathbf{c}_{52}^{T}, \mathbf{c}_{51}^{T}$ . $\mathbf{c}_{99} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{52} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{c}_{83}^{T}, \mathbf{c}_{124}^{T}, \mathbf{c}_{118}^{T}$ . $\mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{124} = \left[\begin{smallmatrix} -3 \\ 6 \\ -15 \end{smallmatrix}\right], \mathbf{c}_{118} = \left[\begin{smallmatrix} 0 \\ -4 \\ 15 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{172}^{T}, \mathbf{c}_{154}^{T}, \mathbf{c}_{218}^{T}$ . $\mathbf{c}_{172} = \left[\begin{smallmatrix} 3 \\ 6 \\ 4 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{154} = \left[\begin{smallmatrix} 6 \\ 13 \\ 5 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{218} = \left[\begin{smallmatrix} 3 \\ -2 \\ 6 \\ 3 \end{smallmatrix}\right]$ .

(5)

$\mathbf{c}_{197}^{T}, \mathbf{c}_{198}^{T}, \mathbf{c}_{207}^{T}$ . $\mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{207} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

(6)

$\mathbf{c}_{255}^{T}, \mathbf{c}_{256}^{T}, \mathbf{c}_{257}^{T}$ . $\mathbf{c}_{255} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{256} = \left[\begin{smallmatrix} 6 \\ 8 \\ 0 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{257} = \left[\begin{smallmatrix} 0 \\ -3 \\ 3 \\ 3 \\ 0 \end{smallmatrix}\right]$ .

Problem 38.6

A $2$ -form is given by a matrix, and a $1$ -form is given by a coordinate row. Find their exterior product.

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(1)

$\mathbf{A}_{254}, \mathbf{c}_{81}^{T}$ . $\mathbf{A}_{254} = \left[\begin{smallmatrix} 0 & 3 & 5 \\ -3 & 0 & 2 \\ -5 & -2 & 0 \end{smallmatrix}\right], \mathbf{c}_{81} = \left[\begin{smallmatrix} 6 \\ 3 \\ 3 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{254}, \mathbf{c}_{66}^{T}$ . $\mathbf{A}_{254} = \left[\begin{smallmatrix} 0 & 3 & 5 \\ -3 & 0 & 2 \\ -5 & -2 & 0 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{252}, \mathbf{c}_{93}^{T}$ . $\mathbf{A}_{252} = \left[\begin{smallmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{93} = \left[\begin{smallmatrix} 1 \\ 2 \\ 2 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{499}, \mathbf{c}_{162}^{T}$ . $\mathbf{A}_{499} = \left[\begin{smallmatrix} 0 & 1 & 1 & 2 \\ -1 & 0 & -1 & 3 \\ -1 & 1 & 0 & 5 \\ -2 & -3 & -5 & 0 \end{smallmatrix}\right], \mathbf{c}_{162} = \left[\begin{smallmatrix} 9 \\ 0 \\ 3 \\ 6 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{432}, \mathbf{c}_{204}^{T}$ . $\mathbf{A}_{432} = \left[\begin{smallmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right], \mathbf{c}_{204} = \left[\begin{smallmatrix} 3 \\ 3 \\ 3 \\ 3 \end{smallmatrix}\right]$ .

Problem 38.7

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

Let $u, u_{1}, v$ and $w$ be exterior forms of degrees $p, p, q$ and $r$ respectively. Prove that:

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(1)

$(\lambda u) \wedge v = \lambda (u\wedge v)$

(2)

$\left(u + u_{1}\right) \wedge v = u\wedge v + u_{1} \wedge v$

(3)

$(u\wedge v) \wedge w = u\wedge (v\wedge w)$

(4)

$u\wedge v = (- 1)^{pq} v\wedge u$ .

Problem 38.8

Prove that the value of a $q$ -form on a system of vectors $x_{1}, \ldots , x_{q}$ actually depends only on the $q$ -vector $x_{1} \wedge \ldots \wedge x_{q}$ .

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Problem 38.9

Let $\mathrm{f}^{1}, \ldots , \mathrm{f}^{p}$ be $1$ -forms. Find the value of the $p$ -form $\mathrm{f}^{1} \wedge \ldots \wedge \mathrm{f}^{p}$ on the system of vectors $x_{1}, \ldots , x_{p}$ .

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Problem 38.10

(1)(2)

A $2$ -form in $\mathcal{L}_{4}$ is given by the row of its essential components $\varphi$ , and the vectors $x$ and $y$ by the coordinate columns $\xi , \eta$ . Find the value of the $2$ -form on the pair $x, y$ :

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(1)

$\varphi = \mathbf{c}_{279}^{T}, \quad \xi = \mathbf{c}_{174}, \quad \eta = \mathbf{c}_{186}$ . $\mathbf{c}_{279} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \\ 1 \\ 2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{174} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{186} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \\ 1 \end{smallmatrix}\right]$ .

(2)

$\varphi = \mathbf{c}_{269}^{T}, \quad \xi = \mathbf{c}_{171}, \quad \eta = \mathbf{c}_{177}$ . $\mathbf{c}_{269} = \left[\begin{smallmatrix} -2 \\ -1 \\ 0 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{177} = \left[\begin{smallmatrix} -2 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

Problem 38.11

Prove that for the linear dependence of the vectors $a_{1}, \ldots , a_{p}$ it is necessary and sufficient that $a_{1} \wedge \ldots \wedge a_{p} = 0$ .

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Problem 38.12

(1)(2)

Let $e_{1}, \ldots , e_{n}$ be a basis in $\mathcal{L}_{n}$ . Prove that:

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(1)

the bivectors $e_{i} \wedge e_{j}$ for all pairs $i, j$ such that $i < j$ form a basis in the space of bivectors of the space $\mathcal{L}_{n}$ .

(2)

the $p$ -vectors $e_{i_{1}} \wedge \ldots \wedge e_{i_{p}}$ for all combinations of indices $i_{1}, \ldots , i_{p}\left(i_{1} < \ldots < i_{p}\right)$ form a basis in the space of $p$ -vectors of the space $\mathcal{L}_{n}$ .

Problem 38.13

To the basis $\mathbf{e} = \left(e_{1}, e_{2}, e_{3}, e_{4}\right)$ of the space $\mathcal{L}_{4}$ assign the basis $\varepsilon = \left(e_{1} \wedge e_{2}, e_{1} \wedge e_{3}, e_{1} \wedge e_{4}, e_{2} \wedge e_{3}, e_{2} \wedge e_{4}, e_{3} \wedge e_{4}\right)$ of the corresponding space of bivectors, and to the basis $\mathbf{e}^{\prime }$ assign the analogously constructed basis $\varepsilon^{\prime }$ . Find the transition matrix from $\varepsilon$ to $\varepsilon^{\prime }$ , if the transition matrix from $e$ to $e^{\prime }$ is $S$ .

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Problem 38.14

The exterior product $u^{i_{1} \ldots i_{n - 1}}$ of the vectors $x_{1}, \ldots \ldots , x_{n - 1}$ from $\mathcal{L}_{n}$ has $n$ essential components. Prove that under a change of basis in $\mathcal{L}_{n}$ with transition matrix $S$ , the row $\mathbf{a} = \left(a^{1}, \ldots , a^{n}\right)$ of essential components $a^{i} = u^{1, \ldots , i - 1, i + 1, \ldots , n}$ transforms by the formula $\mathbf{a}^{\prime } = \mathbf{a} S(\operatorname {det} S)^{ - 1}$ .

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Problem 38.15

Using the result of problem 38. 8, prove that the vector space of $p$ -forms can be identified with the space dual to the vector space of $p$ -vectors.

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Problem 38.16

Prove that in $\mathcal{L}_{3}$ every bivector is decomposable.

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Problem 38.17

Prove that for the decomposability of a bivector $u^{ij}$ in $\mathcal{L}_{4}$ it is necessary and sufficient that the equality $u^{12} u^{34} - u^{13} u^{24} + u^{14} u^{23} = 0$ hold

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Problem 38.18

(1)(2)(3)

Let $a_{1}, a_{2}, a_{3}$ and $a_{4}$ be linearly independent vectors. Are the following bivectors decomposable:

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(1)

$a_{1} \wedge a_{2} + a_{3} \wedge a_{4}$

(2)

$a_{3} \wedge a_{2} + a_{1} \wedge a_{4} + a_{1} \wedge a_{2}$

(3)

$a_{1} \wedge a_{2} + a_{3} \wedge a_{4} + a_{3} \wedge a_{2} + a_{1} \wedge a_{4}$ ?

Problem 38.19

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

Is the bivector in $\mathcal{L}_{4}$ , given in some basis by the column of essential components a, decomposable:

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(1)

$\mathbf{a} = \mathbf{c}_{279}$ . $\mathbf{c}_{279} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \\ 1 \\ 2 \\ 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{a} = \mathbf{c}_{269}$ . $\mathbf{c}_{269} = \left[\begin{smallmatrix} -2 \\ -1 \\ 0 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

(3)

$\mathbf{a} = \mathbf{c}_{278}$ . $\mathbf{c}_{278} = \left[\begin{smallmatrix} -1 \\ 2 \\ 1 \\ 2 \\ -1 \\ 4 \end{smallmatrix}\right]$ .

(4)

$\mathbf{a} = \mathbf{c}_{281}$ ? $\mathbf{c}_{281} = \left[\begin{smallmatrix} 0 \\ 4 \\ 6 \\ 4 \\ 2 \\ 3 \end{smallmatrix}\right]$ .

Problem 38.20

Prove that the subspace generated by a $p$ -vector has dimension $r \leqslant p$ , with equality attained for decomposable $p$ -vectors and only for them.

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Problem 38.21

(1)(2)

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(1)

Let a decomposable bivector in the basis $e_{1}, \ldots , e_{n}$ have components $u^{ij}$ . Prove that the vectors $l^{i} = u^{ij} e_{j}$ lie in the space generated by this bivector.

(2)

State and prove an analogous statement for decomposable $p$ -vectors.

Problem 38.22

Can the dimension of the subspace generated by a bivector in the space $\mathcal{L}_{4}$ equal 1? 3?

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Problem 38.23

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

In some basis of the space $\mathcal{L}_{4}$ , a bivector $u$ is defined by the column of essential components a. Find the linear subspace generated by this bivector:

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(1)

$\mathbf{a} = \mathbf{c}_{279}$ . $\mathbf{c}_{279} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \\ 1 \\ 2 \\ 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{a} = \mathbf{c}_{269}$ . $\mathbf{c}_{269} = \left[\begin{smallmatrix} -2 \\ -1 \\ 0 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

(3)

$\mathbf{a} = \mathbf{c}_{278}$ . $\mathbf{c}_{278} = \left[\begin{smallmatrix} -1 \\ 2 \\ 1 \\ 2 \\ -1 \\ 4 \end{smallmatrix}\right]$ .

(4)

$\mathbf{a} = \mathbf{c}_{280}$ . $\mathbf{c}_{280} = \left[\begin{smallmatrix} -1 \\ 2 \\ 1 \\ 2 \\ -1 \\ 2 \end{smallmatrix}\right]$ .

Problem 38.24

Prove that a decomposable $p$ -vector defining a subspace $\mathcal{L}_{p}$ can be found from this subspace up to a numerical factor.

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Problem 38.25

(1)(2)(3)

A subspace $\mathcal{L}_{2}$ in the space $\mathcal{L}_{4}$ is given by a system of linear equations with matrix $A$ . Find the components of the bivector defining $\mathcal{L}_{2}$ :

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(1)

$A = \mathbf{A}_{502}$ . $\mathbf{A}_{502} = \left[\begin{smallmatrix} 1 & 1 & 7 & -3 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{503}$ . $\mathbf{A}_{503} = \left[\begin{smallmatrix} 3 & -2 & 1 & 1 \\ 2 & -1 & 2 & 0 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{506}$ . $\mathbf{A}_{506} = \left[\begin{smallmatrix} 3 & -4 & 7 & 1 \\ 2 & -3 & 5 & 2 \end{smallmatrix}\right]$ .

Problem 38.26

(1)(2)

Subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ in the space $\mathcal{L}_{4}$ are generated respectively by a vector $x$ and a bivector $u$ . The vector is given by the coordinate column $\xi$ , and the bivector by the column of essential components $\mathbf{a}$ . Verify that $\mathcal{L}_{1} \subset \mathcal{L}_{2}$ , and find a vector $y$ such that $u = x\wedge y$ :

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(1)

$\xi = \left[\begin{smallmatrix} - 2\\ 6\\ 1\\ 1\end{smallmatrix}\right], \mathbf{a} = \left[\begin{smallmatrix} 10\\ 1\\ 3\\ 2\\ - 4\\ - 1\end{smallmatrix}\right]$

(2)

$\xi = \left[\begin{smallmatrix} 1\\ 2\\ 0\\ 1\end{smallmatrix}\right], \mathbf{a} = \left[\begin{smallmatrix} 2\\ 1\\ 3\\ 2\\ 4\\ - 1\end{smallmatrix}\right]$ .

Problem 38.27

(1)(2)

Subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ in the space $\mathcal{L}_{4}$ are generated respectively by a vector $x$ and a bivector $u$ . The vector is given by the coordinate column $\xi$ , and the bivector by the column of essential components $\mathbf{a}$ . Verify that $\mathcal{L}_{1} \cap \mathcal{L}_{2} = \left\{ o\right\}$ , and find a $3$ -vector generating $\mathcal{L} = \mathcal{L}_{1} \oplus \mathcal{L}_{2}$ . Find the equation of the subspace $\mathcal{L}$ :

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(1)

$\xi = \left[\begin{smallmatrix} 2\\ 2\\ 1\\ 1\end{smallmatrix}\right], \mathbf{a} = \left[\begin{smallmatrix} 2\\ 1\\ 3\\ 2\\ 4\\ - 1\end{smallmatrix}\right]$

(2)

$\xi = \left[\begin{smallmatrix} 1\\ 0\\ 1\\ 1\end{smallmatrix}\right], \mathbf{a} = \left[\begin{smallmatrix} 9\\ 5\\ 1\\ 4\\ - 1\\ - 1\end{smallmatrix}\right]$ .

Problem 38.28

Let $1$ -forms $\mathrm{f}^{1}, \ldots , \mathrm{f}^{k}$ be linearly independent, and let the $1$ -forms $g^{1}, \ldots , g^{k}$ satisfy the equality $\mathrm{f}^{1} \wedge g^{1} + \ldots \ldots + \mathrm{f}^{k} \wedge g^{k} = 0$ . Prove that $g^{i} = \sum_{j = 1}^{k} a^{ij} \mathrm{f}^{j}$ for all $i = 1, \ldots , k$ , with $a^{ij} = a^{ji}$ (Cartan's lemma).

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Problem 38.29

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

$1$ -forms $\mathrm{f}^{1}, \mathrm{f}^{2}$ and $g^{1}$ are given by coordinate rows $\varphi^{1}, \varphi^{2}, \xi^{1}$ . Does there exist a $1$ -form $g^{2}$ such that $\mathrm{f}^{1} \wedge g^{1} + \mathrm{f}^{2} \wedge g^{2} = 0$ ? Find all such forms, if they exist:

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(1)

$\varphi^{1} = \mathbf{c}_{172}^{T}, \varphi^{2} = \mathbf{c}_{173}^{T}, \quad \xi^{1} = \mathbf{c}_{166}^{T}$ . $\mathbf{c}_{172} = \left[\begin{smallmatrix} 3 \\ 6 \\ 4 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{173} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(2)

$\varphi^{1} = \mathbf{c}_{197}^{T}, \varphi^{2} = \mathbf{c}_{185}^{T}, \quad \xi^{1} = \mathbf{c}_{166}^{T}$ . $\mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{185} = \left[\begin{smallmatrix} 1 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(3)

$\varphi^{1} = \mathbf{c}_{166}^{T}, \varphi^{2} = \mathbf{c}_{228}^{T}, \quad \xi^{1} = \mathbf{c}_{227}^{T}$ . $\mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{228} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{227} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \\ -1 \end{smallmatrix}\right]$ .

(4)

$\varphi^{1} = \mathbf{c}_{171}^{T}, \varphi^{2} = \mathbf{c}_{186}^{T}, \quad \xi^{1} = \mathbf{c}_{193}^{T}$ . $\mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{186} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{193} = \left[\begin{smallmatrix} 0 \\ 0 \\ 0 \\ 1 \end{smallmatrix}\right]$ .

Problem 38.30

Prove that for every $2$ -form $\omega$ there exists a basis $\mathrm{f}^{1}, \ldots , \mathrm{f}^{n}$ in the space of $1$ -forms such that the form $\omega$ has canonical form $\omega = \mathrm{f}^{1} \wedge \mathrm{f}^{2} + \mathrm{f}^{3} \wedge \mathrm{f}^{4} + \ldots + \mathrm{f}^{2p - 1} \wedge \mathrm{f}^{2p} (2p \leqslant n)$ .

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Problem 38.31

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

A $2$ -form is given by its matrix in some basis. Reduce it to the canonical form described in problem 38. 30:

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(1)

$\mathbf{A}_{250}$ . $\mathbf{A}_{250} = \left[\begin{smallmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{439}$ . $\mathbf{A}_{439} = \left[\begin{smallmatrix} 0 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 \\ -1 & -1 & 0 & 1 \\ -1 & -1 & -1 & 0 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{490}$ . $\mathbf{A}_{490} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{499}$ . $\mathbf{A}_{499} = \left[\begin{smallmatrix} 0 & 1 & 1 & 2 \\ -1 & 0 & -1 & 3 \\ -1 & 1 & 0 & 5 \\ -2 & -3 & -5 & 0 \end{smallmatrix}\right]$ .