Linear Algebra Subsets/Row/Column Spaces

Question

Let T = { v1 = (2 1 1 3), v2 = (3 2 0 1), v3 = (1 0 2 5), v4 = (-1 -1 1 2) v5 = (3 2 0 0), v6 = (13 8 2 8 ) }

(a) Find a linearly independent subset S of T such that span(S) = span(T).

(b) Does T span the whole R^4? If not, find a set T' containing T such that span(T') = R^4.

Thanks

Answer 1

import numpy as np

import sympy

"""

We can approach this problem by using a matrix

which consists of the vectors given in the problem

as the columns, then we use python to find the RREF

of the resulting matrix and its corresponoding rank.

"""

# Defining the vectors from the problem

v1 = np.array([2, 1, 1, 3])

v2 = np.array([3, 2, 0, 1])

v3 = np.array([1, 0, 2, 5])

v4 = np.array([-1, -1, 1, 2])

v5 = np.array([3, 2, 0, 0])

v6 = np.array([13, 8, 2, 8])

# Create a matrix with the vectors as its columns

T = np.array([v1, v2, v3, v4, v5, v6]).T

# Calculate the rank of T to determine the number

# of linearly independent vectors

rank_T = np.linalg.matrix_rank(T)

# Perform row reduction to get the RREF

U, pivots = sympy.Matrix(T).rref()

# Extract the linearly independent vectors from T

# using the pivot columns

S = T[:, pivots]

# Output the results

print("Linearly independent subset S of T:")

print(S)

# Check if T spans R^4

spans_R4 = rank_T == 4

print(f"Does T span R^4? {'Yes' if spans_R4 else 'No'}")

"""

You should get the following output upon running this code:

Linearly independent subset S of T:

[[2 3 3]

[1 2 2]

[1 0 0]

[3 1 0]]

Does T span R^4? No

"""