Linear Algebra: Practice Programs in Python and C Languages
Computation of Dimensions of Subspaces in Vectors by Using Python (Linear Algebra: Vector Spaces)
Linear Algebra: Practice Programs in Python and C Languages - Linear Algebra: Vector Spaces: Computation of Dimensions of Subspaces in Vectors by Using Python
Linear Algebra
PROGRAMMES
IN PYTHON AND C LANGUAGES
UNIT − 1:
VECTOR SPACES
COMPUTATION OF
DIMENSIONS OF SUBSPACES IN VECTORS BY USING PYTHON
To
compute the dimension of a subspace spanned by a set of vectors in Python, we
can use the numpy, linalg.matrix_rank function. The dimensions of the subspace
is equal to the rank of the matrix
formed by the given vectors.
Method: Using numpy.lina1g.matrix_rank
The
most straightforward way is to form a matrix where each row (or column)
represents one of the spanning vectors, and then use the numpy library to
compute its rank. The rank of the matrix is the number of linearly independent
vectors, which is the dimension of the subspace they span.
Here
is the Python code:
python
import numpy as np
def subspace_dimension (vectors):
" " "
Calculates the dimension of the subspace spanned by a list of vector:
Args:
vectors: A list of vectors, where each vector is a list or numpy
Returns:
The dimension of the subspace (integer).
" " "
# Convert the list of vectors into a Numpy matrix.
# The vectors form the rows of the matrix.
vectors_1 = {
[1, 1, 0],
[1, ‒1, 01,
[2, 0, 0]
# This vector is a linear combination of the first two (v)
}
dim_1 = subspace_dimension (vectors_1)
print (f"The dimension of the subspace spanned by vectors 1
is: {dim_1}}
# Another example: A set of vectors that spans all of R3.
vectors_ 2 = [
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]
]
dim_2= subspace_dimension (vectors-2)
print (f"The dimension of the subspace spanned by vectors 2
is: {dim_2}"
# Example of a ID subspace (a line through the origin)
vectors_3 = [
[1, 2, 3],
[2, 4, 6],
[3,6,9]
]
dim_3= subspace_dimension (vectors_3)
print (f"The dimension of the subspace spanned by vectors 3
is: {dim_3}"
Key concepts
Subspace:
A subset of a vector space that is itself a vector space. Subspaces can be
lines, planes, etc., passing through the origin.
Linear Algebra: Practice Programs in Python and C Languages : Tag: maths, mathematics : - Computation of Dimensions of Subspaces in Vectors by Using Python (Linear Algebra: Vector Spaces)
Linear Algebra: Practice Programs in Python and C Languages
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Linear Algebra
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