Linear Algebra: Practice Programs in Python and C Languages
Computation of Null Space and Range of a Matrix by Using Python (Linear Algebra: Linear Transformation and Diagonalization)
Linear Algebra: Practice Programs in Python and C Languages : Linear Algebra: Linear Transformation and Diagonalization: Computation of Null Space and Range of a Matrix by Using Python
Linear Algebra
PROGRAMMES
IN PYTHON AND C LANGUAGES
UNIT − II:
LINEAR TRANSFORMATION AND DIAGONALIZATION
COMPUTATION OF NULL
SPACE AND RANGE OF A MATRIX BY USING PYTHON
The
null space (or kernel) of a matrix A is the set of all vectors x such that
Ax=0. The range (or column space) of a matrix A is the set of all possible
linear combinations of the columns of A. In Python, these can be computed using
libraries like NumPy and SymPy.
1. Null space using SymPy
The
sympy library provides a direct function to compute the null space.
Python
from sympy import matrix
# Define your matrix
A_list = [[1,2,0],
[2,4,0],
[3,6,1]
A = Matrix (A list)
# Compute the null space
null_space_basis = A.nullspace ()
print ("Null Space Basis: ")
for vector in null_space_basis:
print (vector)
# You can verify by multiplying A with a vector from the null
space
# For example, if null_space_basis is not empty, take the first
vector
If null_space_basis:
null_vector = null_space_basis [0]
print ("\nVerification (A * null_vector):")
print (A * null vector)
2. Null Space using NumPy (SVD approach)
NumPy's
scipy.linalg.null_space function, which relies on Singular Value Decomposition
(SVD), can also be used.
Python
import numpy as np
from scipy.linalg import null_space
# Define your matrix
A_np = np.array([1, 2,0],
[2, 4, 0],
[3, 6, 1]])
# Compute the null space
null_space_basis_np = null space (A_np)
print ("Null Space Basis (NumPy): ")
print (null space_basis_np)
# Verification
print("\nVerification (A * null_space_basis_np):")
print (A_np @ null_space_basis_np)
3. Range (Column Space) using NumPy (SVD or RREF)
The
rank space can be found by identifying the linearly independent columns of the
matrix. This can be done by finding the basis for the column space.
Python
import numpy as np
# Define your matrix
A range = np.array([1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
# Using SVD to find a basis for the column space U, S, Vh = np.linalg.svd
(A_range)
#The first 'rank' columns of U form an orthornormal basis for
the column space
rank = np.linalg.matrix_rank (A_range)
range basis svd = U[:, :rank]
print ("Range Space Basis (SVD):")
print (range_basis_svd)
Linear Algebra: Practice Programs in Python and C Languages : Tag: maths, mathematics : - Computation of Null Space and Range of a Matrix by Using Python (Linear Algebra: Linear Transformation and Diagonalization)
Linear Algebra: Practice Programs in Python and C Languages
Under Subject
Linear Algebra
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