Informatics1-2017/Practice11

A MathWikiből
(Változatok közti eltérés)
(What do these do?)
 
130. sor: 130. sor:
 
sum([n for n in range(1, 10) if is_prime(n)])
 
sum([n for n in range(1, 10) if is_prime(n)])
 
</python>
 
</python>
A little spoiler for the following: [https://hu.wikipedia.org/wiki/T%C3%B6k%C3%A9letes_sz%C3%A1mok spoiler]
+
A little spoiler for the following: [https://en.wikipedia.org/wiki/Perfect_number spoiler]
 
<python>
 
<python>
 
[n for n in range(1, 100) if n == sum([m for m in range(1, n) if n % m == 0])]
 
[n for n in range(1, 100) if n == sum([m for m in range(1, n) if n % m == 0])]

A lap jelenlegi, 2017. november 20., 05:41-kori változata

Tartalomjegyzék

Using Sage

Sage server

https://sage.math.bme.hu/

If your browser finds the certificate untrustworthy, accept it manually!


More matrices

Reminder and new stuff

In sage we can define a matrix as follows:

m = matrix([[1, 0], [0, 1]])

This results in the matrix:

1 0
0 1

Blockmatrices, all one matrices, diagonal matrices can be created easily, like in octave:

A = diagonal_matrix([1, 5])
B = ones_matrix(2, 2)
block_matrix([[A, -1*A], [A^(-1), B]])

This results in the matrix:

 1   0| -1   0
 0   5|  0  -5
------+-------
 1   0|  1   1
 0 1/5|  1   1

The det method calculates the determinant of the matrix:

m.det()


Tasks

Blockmatrix

Calculate the determinant of the following blockmatrix:

X I
O X

where I is the 3x3 identity matrix and O is the 3x3 all 0 matrix, X is the following:

 0 -1 -1
-1  0 -1
-1 -1  0

Equations

Solve the following system of equations with the form Ax = b, where A and b are:

 1 -1  0  |  1
 3  1 -1  |  1
-2  0  1  |  2

Use the solve_right method from the lecture!

Once you have the solution, make the matrix into a matrix over the ring GF(3) (with the change_ring method), solve it with this new matrix.

Linear independence

Find out for which values of x would the rows (or columns) of the following matrix be dependent / independent. (Use the solve method.)

x  0  1
0  2  x
1  x -1



List comprehension

Reminder

[expression for element in iterable_thing]

This creates a list which contains the expression for every element of iterable_thing. An iterable thing is a list for example, like a list created with the range function.

[expression if condition for element in iterable_thing]

Similar to the previous one, except only the elements for which the condition holds will be included.

[expression if condition1 else expression_alt for element in iterable_thing1
           for element2 in iterable_thing2
           ...
           for elementN in iterable_thingN]

We can write multiple fors.

Example:

[n^2 for n in range(1, 5)]  # [1, 4, 9, 16]
[n for n in [-1, 2, -3, 4] if n > 0]  # [2, 4]

Tasks

What do these do?

Execute the following, and then study how they work.

[n for n in range(1, 10)]
[(n, m) for n in range(1, 10) for m in range(1, 5)]
[n for n in range(1, 10) if is_prime(n)]
[n for n in range(1, 100) if n % 5 == 0 and n % 7 == 1]
[(n, m) for n in range(1, 5) for m in range(n, 5)]
[(m, n) for n in range(1, 10) for m in range(n, 10) if m % n == 0]
sorted([(m, n) for n in range(1, 10) for m in range(n, 10) if m % n == 0])
sum([n for n in range(1, 10) if is_prime(n)])

A little spoiler for the following: spoiler

[n for n in range(1, 100) if n == sum([m for m in range(1, n) if n % m == 0])]

Solve the following

  1. Find the square numbers x below 1000, where x + 1 is a prime. For example 4. (Find all of these numbers.)
  2. Find those (x, y) pairs of numbers, where both are prime and their integer division (//) is prime as well. For example (11, 2).
  3. Find the 3 digit numbers with the form xyz, where xyz (= 100 * x + 10 * y + z) = x^3 + y^3 + z^3. For example 1, 5, 3, because 1^3 + 5^3 + 3^3 == 153.
  4. Find the numbers below 10000, which can be written as the sum of 2 cube numbers in two different ways.
Személyes eszközök