numbers.add(5)
print(numbers) # Outputs: {1, 2, 3, 4, 5}
4.3 Removing Elements from a Set
Sets in Python allow for the removal of elements using methods like remove(),
discard(), and pop(). The difference is that remove() raises an error
if the element doesn't exist, while discard() does not.
my_set = {1, 2, 3, 4, 5}
my_set.remove(3) # Removes the element 3
print(my_set) # Outputs: {1, 2, 4, 5}
my_set.discard(5) # Discards the element 5
print(my_set) # Outputs: {1, 2, 4}
# Using pop to remove a random element
removed_element = my_set.pop()
print(f"Removed: {removed_element}") # Outputs: Removed: 1 (or another element)
print(my_set) # Remaining elements
4.4 Python Set Operations
Python sets support common mathematical operations like union,
intersection, difference, and symmetric difference. These can be performed using
operators or set methods:
Union (|): Combines two sets, removing duplicates.Intersection (&): Finds common elements between two sets.Difference (-): Finds elements present in the first set but not the second.Symmetric Difference (^): Finds elements present in either set but not both.
A = {1, 2, 3, 4}
B = {3, 4, 5, 6}
# Union
print(A | B) # Outputs: {1, 2, 3, 4, 5, 6}
# Intersection
print(A & B) # Outputs: {3, 4}
# Difference
print(A - B) # Outputs: {1, 2}
# Symmetric Difference
print(A ^ B) # Outputs: {1, 2, 5, 6}
4.5 Built-in Functions with Set
Python provides various built-in functions that work with sets:
len(): Returns the number of elements in a set.max() and min(): Returns the largest and smallest elements, respectively.sum(): Returns the sum of elements in a set (for numeric sets).sorted(): Returns a sorted list of the set’s elements.
my_set = {7, 2, 10, 4}
print(len(my_set)) # Outputs: 4
print(max(my_set)) # Outputs: 10
print(min(my_set)) # Outputs: 2
print(sum(my_set)) # Outputs: 23
print(sorted(my_set)) # Outputs: [2, 4, 7, 10]
Assignment 5: Python Dictionary
5.1 Creating a Dictionary
Dictionaries store key-value pairs in Python. You can create a dictionary using
curly braces {} and colons to separate keys from values.
my_dict = {"name": "Alice", "age": 25, "city": "New York"}
print(my_dict) # Outputs: {'name': 'Alice', 'age': 25, 'city': 'New York'}
5.2 Changing a Dictionary in Python
Dictionaries are mutable, meaning you can add, modify, or remove key-value pairs after the dictionary has been created.
# Adding or updating a key-value pair
my_dict["age"] = 30
my_dict["country"] = "USA"
print(my_dict) # Outputs: {'name': 'Alice', 'age': 30, 'city': 'New York', 'country': 'USA'}
5.3 Removing Elements from a Dictionary
You can remove elements from a dictionary using methods like del, pop(), and clear():
my_dict = {"name": "Alice", "age": 25, "city": "New York"}
# Removing a specific key-value pair
del my_dict["age"]
print(my_dict) # Outputs: {'name': 'Alice', 'city': 'New York'}
# Using pop to remove a key and return its value
city = my_dict.pop("city")
print(city) # Outputs: New York
print(my_dict) # Outputs: {'name': 'Alice'}
5.4 Python Dictionary Operations
keys(): Returns a view object of all the keys in the dictionary.values(): Returns a view object of all the values.items(): Returns a view object of all key-value pairs.
my_dict = {"name": "Alice", "age": 25, "city": "New York"}
# Getting keys, values, and items
print(my_dict.keys()) # Outputs: dict_keys(['name', 'age', 'city'])
print(my_dict.values()) # Outputs: dict_values(['Alice', 25, 'New York'])
print(my_dict.items()) # Outputs: dict_items([('name', 'Alice'), ('age', 25), ('city', 'New York')])
5.5 Built-in Functions with Dictionary
Some useful built-in functions that work with dictionaries include:
len(): Returns the number of key-value pairs in the dictionary.dict(): Creates a dictionary from another mapping type.zip(): Combines two sequences into a dictionary.
names = ["Alice", "Bob", "Charlie"]
ages = [25, 30, 35]
people = dict(zip(names, ages))
print(people) # Outputs: {'Alice': 25, 'Bob': 30, 'Charlie': 35}
Assignment 6: Decision Making Statements
6.1 IF statement
The if statement is used to execute a block of code only if a certain condition is true.
x = 10
if x > 5:
print("x is greater than 5") # This will be executed
6.2 IF...ELIF...ELSE Statements
If you want to test multiple conditions, you can use elif (else if) and else blocks.
x = 10
if x > 15:
print("x is greater than 15")
elif x > 5:
print("x is greater than 5 but less than or equal to 15") # This will be executed
else:
print("x is less than or equal to 5")
6.3 Nested IF Statements
It's possible to nest if statements inside one another to check multiple conditions.
x = 10
y = 20
if x > 5:
if y > 15:
print("x is greater than 5 and y is greater than 15") # This will be executed
6.4 While Loop
A while loop repeatedly executes a block of code as long as a condition is true.
i = 1
while i <= 5:
print(i)
i += 1 # Increment i
6.5 For Loop
A for loop is used to iterate over a sequence (like a list, tuple, or string).
for i in range(5):
print(i)
Assignment 7: Using SymPy for Basic Operations on Matrices
7.1 Addition, Subtraction, Multiplication, Power
SymPy allows for basic matrix operations like addition, subtraction, multiplication, and exponentiation.
from sympy import Matrix
# Creating matrices
A = Matrix([[1, 2], [3, 4]])
B = Matrix([[5, 6], [7, 8]])
# Addition
C = A + B
print(C) # Outputs the matrix result of A + B
# Subtraction
D = A - B
print(D) # Outputs the matrix result of A - B
# Multiplication
E = A * B
print(E) # Outputs the matrix result of A * B
# Power
F = A ** 2
print(F) # Outputs the matrix result of A squared
7.2 Accessing Elements, Rows, Columns of a Matrix
Individual elements, rows, and columns of a matrix can be accessed using SymPy’s indexing methods.
element = A[0, 1] # Access element at row 0, column 1
print(element) # Outputs: 2
row = A.row(1) # Access row 1
print(row) # Outputs: [3, 4]
column = A.col(0) # Access column 0
print(column) # Outputs: [1, 3]
7.3 Creating Some Standard Matrices
SymPy provides functions for creating standard matrices such as
zero matrices, identity matrices, and diagonal matrices.
from sympy import zeros, eye, diag
zero_matrix = zeros(2, 3) # 2x3 zero matrix
identity_matrix = eye(3) # 3x3 identity matrix
diagonal_matrix = diag(1, 2, 3) # Diagonal matrix
print(zero_matrix)
print(identity_matrix)
print(diagonal_matrix)
Assignment 8: Using SymPy for Operations on Matrices
8.1 Inserting an Element into a Matrix
In SymPy, although matrices are immutable, you can create a new matrix by
replacing elements or inserting them. To insert a new element into a row or
column, you may have to rebuild the matrix using concatenation or element
replacement methods.
from sympy import Matrix
# Create a matrix
A = Matrix([[1, 2], [3, 4]])
# Rebuilding the matrix by inserting an element at position (1, 1)
A_new = A.row_insert(1, Matrix([[5, 6]]))
print(A_new) # Outputs the new matrix with inserted row
8.2 Inserting a Matrix into Another Matrix
You can insert a submatrix into another matrix by using methods
like row_insert or col_insert for adding rows or columns.
from sympy import Matrix
A = Matrix([[1, 2], [3, 4]])
B = Matrix([[5, 6]])
# Insert matrix B as a new row in matrix A
A_new = A.row_insert(1, B)
print(A_new) # Outputs the new matrix
8.3 Deleting a Row or Column from a Matrix
You can delete a row or column from a matrix by creating a new matrix without that row or column.
from sympy import Matrix
A = Matrix([[1, 2], [3, 4], [5, 6]])
# Delete the second row
A_new = A.row_del(1)
print(A_new) # Outputs the matrix after deleting row 1
8.4 Elementary Row Operations
Elementary row operations include row swapping, row multiplication,
and row addition. These operations are useful for manipulating matrices,
especially in solving systems of equations.
from sympy import Matrix
A = Matrix([[1, 2], [3, 4]])
# Swap row 0 and row 1
A_new = A.elementary_row_op('n<->m', n=0, m=1)
print(A_new) # Outputs: Matrix([[3, 4], [1, 2]])
# Multiply row 0 by 2
A_new = A.elementary_row_op('n->kn', n=0, k=2)
print(A_new) # Outputs: Matrix([[2, 4], [3, 4]])
# Add row 0 to row 1
A_new = A.elementary_row_op('n->n+km', n=1, m=0, k=1)
print(A_new) # Outputs: Matrix([[1, 2], [4, 6]])
Assignment 9: Using SymPy to Obtain Matrix Properties
9.1 Determinant of a Matrix
The determinant of a matrix is a scalar value that can be computed using
the det() method. It is crucial in solving linear systems,
finding the inverse of matrices, and understanding matrix properties.
from sympy import Matrix
A = Matrix([[1, 2], [3, 4]])
# Calculate determinant
det_A = A.det()
print(det_A) # Outputs: -2
9.2 Rank of a Matrix
The rank of a matrix is the maximum number of linearly independent row
or column vectors. You can compute it using the rank() method in SymPy.
rank_A = A.rank()
print(rank_A) # Outputs: 2
9.3 Transpose of a Matrix
The transpose of a matrix is achieved by swapping its rows and columns.
This operation is useful in many mathematical contexts.
transpose_A = A.T
print(transpose_A) # Outputs the transpose of matrix A
9.4 Reduced Row Echelon Form of a Matrix
The reduced row echelon form (RREF) of a matrix is used in solving systems of
linear equations. It transforms the matrix into a form where back substitution
can be applied easily.
rref_A, pivots = A.rref()
print(rref_A) # Outputs the RREF of matrix A
Assignment 10: Using SymPy for Matrix Inverses and Related Operations
10.1 The Inverse of a Matrix
You can calculate the inverse of a matrix using the inv() method.
The matrix must be square and have a non-zero determinant to be invertible.
inverse_A = A.inv()
print(inverse_A) # Outputs the inverse of matrix A
10.2 Inverse of a Matrix by Row Reduction
The inverse of a matrix can also be computed using row reduction techniques,
where elementary row operations are applied until the matrix is reduced to the identity matrix.
# SymPy allows you to apply row reduction techniques to find the inverse
inverse_A = A.inv(method="GE")
print(inverse_A)
10.3 Minor and Cofactors of a Matrix
The minor of an element is the determinant of the matrix obtained by deleting
the element's row and column. The cofactor includes the minor along with a sign.
# Minor of element at (0, 0)
minor_A = A.minor_submatrix(0, 0).det()
print(minor_A) # Outputs the minor of element A[0, 0]
10.4 Inverse of a Matrix by Adjoint Method
The adjoint method is another way to compute the inverse of a matrix.
It involves computing the cofactor matrix, transposing it, and dividing
by the determinant.
adjoint_A = A.adjugate()
inverse_A = adjoint_A / A.det()
print(inverse_A) # Outputs the inverse using the adjoint method
Assignment 11: Using SymPy for LU Decomposition
11.1 Lower Triangular Matrix
A lower triangular matrix is a matrix where all the entries above the
diagonal are zero. SymPy can compute this using LU decomposition.
L, U, _ = A.LUdecomposition()
print(L) # Outputs the lower triangular matrix L
11.2 Upper Triangular Matrix
An upper triangular matrix is one where all the entries below the diagonal
are zero. SymPy provides this using LU decomposition as well.
print(U) # Outputs the upper triangular matrix U
11.3 LU Decomposition of a Matrix
LU decomposition expresses a matrix as the product of a lower triangular matrix
and an upper triangular matrix. This is useful in solving linear systems and computing matrix inverses.
# The matrices L and U are already computed in the LUdecomposition step
print(L * U) # Rebuilds the original matrix A
Assignment 12: Using SymPy to Solve Systems of Linear Equations
12.1 Cramer's Rule
Cramer’s rule is used to solve a system of linear equations using determinants.
It works for systems where the number of equations equals the number of unknowns.
from sympy import symbols
x, y = symbols('x y')
eq1 = 2*x + y - 1
eq2 = x - 2*y - 3
sol = A.solve((x, y))
print(sol) # Outputs the solution using Cramer's rule
12.2 Gauss Elimination Method
Gauss elimination transforms a system of linear equations into an
upper triangular matrix, which is then solved using back substitution.
# Use row reduction (RREF) to solve the system
rref_A, pivots = A.rref()
print(rref_A)
12.3 Gauss-Jordan Method
Gauss-Jordan elimination extends the Gauss method by reducing the matrix to its
reduced row echelon form (RREF), allowing direct solutions without back substitution.
rref_A, pivots = A.rref()
print(rref_A) # Outputs the RREF, which can be directly solved
12.4 LU Decomposition Method for Solving Systems
LU decomposition can also be used to solve systems of linear equations
by solving two triangular systems: one for the lower
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