Exercises and solutions, linear algebra

Determinant

Exercise 1: Simple matrix

Compute the determinant

Solution: Simple matrix

import numpy as np 

# Defining the matrix
A = np.array([[1, 3, 2],
            [4, 1, 3],
            [2, 5, 2]])

det = np.linalg.det(A)
print(det)


# Output:
16.999999999999993   # e.i. det(A) = 17

Exercise 2: Complex numbers

This example is taken from an example exam in TMA4110 - Calculus 3, check out the exam and the solution for hand calculations (only in norwegain). This is task 4.

Comput the determinant

Solution: Complex Numbers

import numpy as np 

# Defining the matrix
# Notice how we can define complex numbers in python by adding the letter "j"
# "j" is used to denote an imaginary unit instead of "i", because Python follows engineering,
# where "i" denotes the electric current as a function of time (especially electrial engineering)
A = np.array([[3, 1-1j, 1j, 4],
            [3, 1, 1-2j, 4+7j],
            [6j, 2+2j, -2, 3j],
            [-3, -1+1j, 1, 3-4j]], dtype=complex)

det = np.linalg.det(A)
print(det)


# Output:
(-15-15j)   # e.i. det(A) = -15-15i

Least squares solution to a linear system

Exercise 1:

This example is taken from an example exam in TMA4110 - Calculus 3, check out the exam and the solution for hand calculations (only in norwegain). This is task 5.

Use the least squares method to find an approximation to the linear system

First, lets write the equations as matrices on the format

, whereTo solve the problem in NumPy, we will use the function numpy.linalg.lstsq which is currently not described in Matrices and linear algebra. We recommend you to read the function documentation before proceeding.

Solution

import numpy as np 

# Defining the matrices
A = np.array([[1, 2],
            [3, 4],
            [5, 6]])
b = np.array([[1],[2],[3]])

x, residuals, rank, s = np.linalg.lstsq(A, b, rcond=None)   
# "rcond=None" sets new default for rcond, see function documentation

print(x)    # "x" is the solution


# Output:
[[-5.97106181e-17]
 [ 5.00000000e-01]]

We can interpret this such as the least squares solution is

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