CoCalc -- ps1-zadania.ipynb

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ubuntu2204

Kernel: SageMath 10.1

Lista I

Zadanie 1.1

Wykonaj poniższe rachunki:

\begin{equation*} \begin{bmatrix} 2 & -1 & 3 \\ 1 & 0 & 2 \end{bmatrix} + 2 \begin{bmatrix} 1 & -2 & 0 \\ -3 & 4 & -1 \end{bmatrix} \end{equation*}

\begin{equation*} \left( 2 \begin{bmatrix} 0 & -2 & 1 \\ 2 & -3 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 3 \\ 2 & -3 \\ 1 & 2 \end{bmatrix}^{\top} \right) \begin{bmatrix} -1 & 0 & 0 \\ 1 & -2 & 0 \\ 2 & 1 & 0 \end{bmatrix}^{\top} -3 \begin{bmatrix} 1 & -1 & 0 \\ 0 & -2 & 1 \end{bmatrix} \end{equation*}

In [15]:

A=matrix(3,4,[3,5,6,0,-1,-4,8,0,-3,0,-2,3])
B=matrix(3,3,[0,-10,-7,-3,2,-1,7,-7,6])
C=matrix(3,3,[1,8,2,5,0,-1,1,1,2])
show(A,B,C)

Out[15]:

$\displaystyle \left(\begin{array}{rrrr} 3 & 5 & 6 & 0 \\ -1 & -4 & 8 & 0 \\ -3 & 0 & -2 & 3 \end{array}\right) \left(\begin{array}{rrr} 0 & -10 & -7 \\ -3 & 2 & -1 \\ 7 & -7 & 6 \end{array}\right) \left(\begin{array}{rrr} 1 & 8 & 2 \\ 5 & 0 & -1 \\ 1 & 1 & 2 \end{array}\right)$

In [22]:

C=C^3
D=5*B*C
Xt=

Out[22]:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [22], line 5
      3 F=D*E
      4 G=transpose(A)
----> 5 H=F*G
      6 show(D,E,F,G)
File /ext/sage/10.1/src/sage/structure/element.pyx:3939, in sage.structure.element.Matrix.__mul__()
   3937 
   3938         if BOTH_ARE_ELEMENT(cl):
-> 3939             return coercion_model.bin_op(left, right, mul)
   3940 
   3941         cdef long value
File /ext/sage/10.1/src/sage/structure/coerce.pyx:1269, in sage.structure.coerce.CoercionModel.bin_op()
   1267     # We should really include the underlying error.
   1268     # This causes so much headache.
-> 1269     raise bin_op_exception(op, x, y)
   1270 
   1271 cpdef canonical_coercion(self, x, y):
TypeError: unsupported operand parent(s) for *: 'Full MatrixSpace of 3 by 3 dense matrices over Integer Ring' and 'Full MatrixSpace of 4 by 3 dense matrices over Integer Ring'

In [21]:

H=F*G
show(H)

Out[21]:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [21], line 1
----> 1 H=F*G
      2 show(H)
File /ext/sage/10.1/src/sage/structure/element.pyx:3939, in sage.structure.element.Matrix.__mul__()
   3937 
   3938         if BOTH_ARE_ELEMENT(cl):
-> 3939             return coercion_model.bin_op(left, right, mul)
   3940 
   3941         cdef long value
File /ext/sage/10.1/src/sage/structure/coerce.pyx:1269, in sage.structure.coerce.CoercionModel.bin_op()
   1267     # We should really include the underlying error.
   1268     # This causes so much headache.
-> 1269     raise bin_op_exception(op, x, y)
   1270 
   1271 cpdef canonical_coercion(self, x, y):
TypeError: unsupported operand parent(s) for *: 'Full MatrixSpace of 3 by 3 dense matrices over Integer Ring' and 'Full MatrixSpace of 4 by 3 dense matrices over Integer Ring'

Zadanie 2.1

Znajdź macierz $X$ spełniajacą poniżesze równanie

\begin{equation*} 2X - \begin{bmatrix} -2 & 2 & -1 \\ 4 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 & -3 \\ 2 & 1 & 3 \end{bmatrix} \end{equation*}

In [37]:

A=matrix(2,3,[-2,2,-1,4,0,1])
B=matrix(2,3,[0,1,-3,2,1,3])
C=B+A
X=C/2
show(C,X)

Out[37]:

$\displaystyle \left(\begin{array}{rrr} -2 & 3 & -4 \\ 6 & 1 & 4 \end{array}\right) \left(\begin{array}{rrr} -1 & \frac{3}{2} & -2 \\ 3 & \frac{1}{2} & 2 \end{array}\right)$

Zadanie 3.1

Wyznaczyć rząd macierzy $A$

a. $\begin{equation*} A = \begin{bmatrix} 5 & 28 & 1 & 4 & -5 & 1 & 0 \\ 0 & 0 & -3 & 0 & -1 & 1 & -2 \\ 41 & 1 & 1 & 0 & -1 & 0 & 0 \\ 1 & 0 & 4 & -1 & 2 & -11 & 1 \\ 0 & 0 & -1 & 4 & -6 & -2 & 0 \\ 1 & -1 & 13 & 1 & -2 & 1 & 1 \\ -3 & -5 & 0 & -1 & 1 & -1 & 2 \end{bmatrix} \end{equation*}$

Zadanie 4.1

Znajdź rozwiązanie układu:

\begin{bmatrix} 1 & 11 & 9 \\ 2 & 8 & 7 \\ 6 & 5 & 4 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ 10 \\ 1 \end{bmatrix}

In [53]:

X = matrix(4,4,[1, 2, 10, 11,1/2, 3, 11, -1,-1, 1, -4, 12,14, 22, -3, 3])
Y = matrix(QQ, [[1, 2, 10], [2, 4, 20], [14, 22, -3]])
Z = matrix(QQ, [[1], [2], [-1]])
show(X,Y,Z)

Out[53]:

$\displaystyle \left(\begin{array}{rrrr} 1 & 2 & 10 & 11 \\ \frac{1}{2} & 3 & 11 & -1 \\ -1 & 1 & -4 & 12 \\ 14 & 22 & -3 & 3 \end{array}\right) \left(\begin{array}{rrr} 1 & 2 & 10 \\ 2 & 4 & 20 \\ 14 & 22 & -3 \end{array}\right) \left(\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right)$

In [61]:

A=X.inverse()
B=X**2
U = Y.augment(Z, subdivide=True)
show(U)

Out[61]:

$\displaystyle \left(\begin{array}{rrr|r} 1 & 2 & 10 & 1 \\ 2 & 4 & 20 & 2 \\ 14 & 22 & -3 & -1 \end{array}\right)$

Zadanie 4.2

Z podanego układu równań utwórz macierz uzupełnioną, sprowadź macierz uzupełnioną $U$ do postaci schodkowej. Znajdź rozwiązanie układu dowolną metodą.

a. $\begin{equation*} \begin{cases} 3 x_{1} + 2 x_{3} -3 x_{4} = 2 \\ x_{2} + 2 x_{4} + x_{5} = 0 \\ - x_{1} + x_{3} + 2 x_{5} = 3 \end{cases} \end{equation*}$

In [0]:

Zadanie 4.3

Rozwiąż układ równań, podaj i narysuj interpretację geometryczną rozwiązania.

i. $$ x $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$

y $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$ = $\begin{bmatrix} 0 \\ 3 \end{bmatrix}$ $$

In [0]:

Zadanie 5.1

Wykonaj następujące rachunki w odpowiednim ciele lub pierścieniu.

a. $3 + 4 \pmod{5}$ b. $2 \cdot 4 \pmod{5}$ c. $3 + \left( 2 \cdot 4 \right) \pmod{5}$ d. $41^{1024} \pmod{13}$

In [0]:

Zadanie 6.1

Oblicz iloczyn permutacji:

\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 3 & 1 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 2 & 1 & 3 \end{pmatrix}

In [0]:

Zadanie 6

Wyznacz permutację odwrotną do permutacji $\sigma$ :

\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{pmatrix}

In [0]:

Zadanie 7

Określ parzystość permutacji $\sigma$

a. $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 5 & 2 & 1 & 6 & 4 & 8 & 7 \end{pmatrix}$

In [0]:

Zadanie 8

Pokaż tabelkę dla wskazanego działania oraz obiektu. Sprawdź czy działanie jest przemienne.

i. $\left( \mathbb{Z}_{6}, + \right)$ ii. $\left( \mathbb{Z}_{6}, \cdot \right)$ iii. $\LaTeX$

In [0]:

Zadanie A

Wykonaj obliczenia dla algorytmu PageRank dla zadania z wykładu 1.

In [3]:

a=matrix(3,2,[0,-4,2,4,-6,0])
b=matrix(2,3,[1,3,2,-2,1,2])
c=matrix(3,3,[-1,1,2,0,-2,1,0,0,0])
d=matrix(2,3,[3,-3,0,0,6,3])

show(a+b)

Out[3]:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [3], line 6
      3 c=matrix(Integer(3),Integer(3),[-Integer(1),Integer(1),Integer(2),Integer(0),-Integer(2),Integer(1),Integer(0),Integer(0),Integer(0)])
      4 d=matrix(Integer(2),Integer(3),[Integer(3),-Integer(3),Integer(0),Integer(0),Integer(6),Integer(3)])
----> 6 show(a+b)
File /ext/sage/10.1/src/sage/structure/element.pyx:1226, in sage.structure.element.Element.__add__()
   1224 # Left and right are Sage elements => use coercion model
   1225 if BOTH_ARE_ELEMENT(cl):
-> 1226     return coercion_model.bin_op(left, right, add)
   1227 
   1228 cdef long value
File /ext/sage/10.1/src/sage/structure/coerce.pyx:1269, in sage.structure.coerce.CoercionModel.bin_op()
   1267     # We should really include the underlying error.
   1268     # This causes so much headache.
-> 1269     raise bin_op_exception(op, x, y)
   1270 
   1271 cpdef canonical_coercion(self, x, y):
TypeError: unsupported operand parent(s) for +: 'Full MatrixSpace of 3 by 2 dense matrices over Integer Ring' and 'Full MatrixSpace of 2 by 3 dense matrices over Integer Ring'