Informatics1-2017/Practice11
A MathWikiből
(Változatok közti eltérés)
(Új oldal, tartalma: „== Using Sage == === Sage server === https://sage.math.bme.hu/ If your browser finds the certificate untrustworthy, accept it manually! = More matrices = == Remin…”) |
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| 74. sor: | 74. sor: | ||
| − | = | + | = List comprehension = |
| − | == | + | == Reminder == |
<python> | <python> | ||
| − | [ | + | [expression for element in iterable_thing] |
</python> | </python> | ||
| − | + | 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. | |
<python> | <python> | ||
| − | [ | + | [expression if condition for element in iterable_thing] |
</python> | </python> | ||
| − | + | Similar to the previous one, except only the elements for which the '''condition''' holds will be included. | |
<python> | <python> | ||
| − | [ | + | [expression if condition1 else expression_alt for element in iterable_thing1 |
| − | for | + | for element2 in iterable_thing2 |
| − | for | + | ... |
| + | for elementN in iterable_thingN] | ||
</python> | </python> | ||
| − | + | We can write multiple fors. | |
| − | + | Example: | |
<python> | <python> | ||
[n^2 for n in range(1, 5)] # [1, 4, 9, 16] | [n^2 for n in range(1, 5)] # [1, 4, 9, 16] | ||
| 99. sor: | 100. sor: | ||
</python> | </python> | ||
| − | == | + | == Tasks == |
| − | === | + | === What do these do? === |
| − | + | Execute the following, and then study how they work. | |
<python> | <python> | ||
| 129. 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] | |
<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])] | ||
</python> | </python> | ||
| − | === | + | === Solve the following === |
| − | + | # Find the square numbers x below 1000, where x + 1 is a prime. For example 4. (Find all of these numbers.) | |
| − | # | + | # Find those (x, y) pairs of numbers, where both are prime and their integer division (//) is prime as well. For example (11, 2). |
| − | # | + | # 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. |
| − | # | + | # Find the numbers below 10000, which can be written as the sum of 2 cube numbers in two different ways. |
| − | + | ||
| − | # | + | |
A lap 2017. november 20., 06:40-kori változata
Tartalomjegyzék |
Using Sage
Sage server
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
- Find the square numbers x below 1000, where x + 1 is a prime. For example 4. (Find all of these numbers.)
- Find those (x, y) pairs of numbers, where both are prime and their integer division (//) is prime as well. For example (11, 2).
- 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.
- Find the numbers below 10000, which can be written as the sum of 2 cube numbers in two different ways.
