diff options
| -rw-r--r-- | 1.py | 26 | ||||
| -rw-r--r-- | libsolve.py | 13 | ||||
| -rw-r--r-- | phw2.tex | 54 |
3 files changed, 91 insertions, 2 deletions
@@ -0,0 +1,26 @@ +from libsolve import Permutation +from itertools import permutations + +PRINT_LATEX = False + +# (1 2 5 8 3 7 6)(4) +rhs = Permutation([1, 4, 6, 3, 7, 0, 5, 2]) + +# (2 8 5 7 3 6 4)(1) +lhs = Permutation([0, 7, 5, 1, 6, 3, 2, 4]) + +for perm in permutations(range(8), 8): + p = Permutation(perm) + + id = list(range(8)) + right = rhs.apply(id) + left = p.apply(lhs.apply(p.apply(id))) + if left == right: + if PRINT_LATEX: + print('\\begin{pmatrix}', end='') + print(*range(1, 9), sep=' & ', end='') + print(*map(lambda i: f'{i + 1}', p.perm), + sep=' & ', + end='\\end{pmatrix}\\\\\n') + else: + print(p.perm) diff --git a/libsolve.py b/libsolve.py index 61d7f40..af72264 100644 --- a/libsolve.py +++ b/libsolve.py @@ -267,3 +267,16 @@ class Matrix: row.set_hints(max_hints) return '\n'.join(repr(row) for row in self.rows) + + +class Permutation: + def __init__(self, perm: Iterable): + self.perm = list(perm) + + def apply(self, now: list): + assert len(self.perm) == len(now) + res = [0] * len(self.perm) + for i in range(len(self.perm)): + res[self.perm[i]] = now[i] + + return res @@ -3,6 +3,7 @@ %\usepackage[utf8]{inputenc} %\usepackage[russian]{babel} +\usepackage{datetime2} \input{intro} @@ -21,15 +22,64 @@ \title{ИДЗ-2 по линейной алгебре} \author{Шарафатдинов Камиль БПМИ-192} -\date{\today} +\date{\DTMnow} % -- Here bet dragons -- \begin{document} + \maketitle +\begin{center} +\url{https://git.fcked.net/alg2} +\end{center} %\section*{Abstract} \clearpage +\question{1}{ + \[ + \sigma(2\ 8\ 5\ 7\ 3\ 6\ 4)\sigma = \left( + \begin{pmatrix} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ + 3 & 7 & 2 & 4 & 6 & 8 & 5 & 1 + \end{pmatrix}^{-1} + \begin{pmatrix} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ + 4 & 6 & 8 & 7 & 3 & 1 & 2 & 5 + \end{pmatrix}^{15} + \right)^{180} + \] +} + + \begin{gather*} + \begin{pmatrix} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ + 3 & 7 & 2 & 4 & 6 & 8 & 5 & 1 + \end{pmatrix}^{-1} = + \begin{pmatrix} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ + 8 & 3 & 1 & 4 & 7 & 5 & 2 & 6 + \end{pmatrix} = (1\ 8\ 6\ 5\ 7\ 2\ 3)(4)\\ + \begin{pmatrix} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ + 4 & 6 & 8 & 7 & 3 & 1 & 2 & 5 + \end{pmatrix}^{15} = (1\ 4\ 7\ 2\ 6)(3\ 8\ 5)^{15} = id_8\\\\ + ((1\ 8\ 6\ 5\ 7\ 2\ 3)(4))^{180} = + ((1\ 8\ 6\ 5\ 7\ 2\ 3)(4))^5 = + (1\ 2\ 5\ 8\ 3\ 7\ 6)(4) + \end{gather*} + + Теперь просто переберем (\texttt{1.py}) все перестановки и выпишем подходящие: + \begin{gather*} + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\3 & 6 & 7 & 5 & 8 & 2 & 4 & 1\end{pmatrix}, + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\3 & 6 & 8 & 5 & 7 & 2 & 4 & 1\end{pmatrix}\\ + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\4 & 2 & 7 & 1 & 8 & 6 & 3 & 5\end{pmatrix}, + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\4 & 2 & 8 & 1 & 7 & 6 & 3 & 5\end{pmatrix}\\ + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\5 & 1 & 7 & 3 & 8 & 4 & 2 & 6\end{pmatrix}, + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\5 & 1 & 8 & 3 & 7 & 4 & 2 & 6\end{pmatrix}\\ + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\6 & 3 & 7 & 2 & 8 & 5 & 1 & 4\end{pmatrix}, + \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\6 & 3 & 8 & 2 & 7 & 5 & 1 & 4\end{pmatrix}\\ + \end{gather*} + \question{2}{ \[ \begin{pmatrix} @@ -85,7 +135,7 @@ -1 & -2 & 2 & -2 & 0 & 1 & 0 & 0\\ 1 & 2 & -1 & 3 & 0 & 0 & 1 & 0\\ -1 & -2 & 1 & -2 & 0 & 0 & 0 & 1 - \end{pmatrix} \overset{\texttt{main2.py}}\sim % TODO + \end{pmatrix} \overset{\texttt{2.py}}\sim % TODO \begin{pmatrix}[cccc|cccc] 1 & 0 & 0 & 0 & 2 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & -1 & 0 & -1 & -2\\ |