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
, where
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]]