numpy as np from sympy import Matrix

Define the vectors as rows of a matrix

vectors = np.array([ [1, 0, 1, -2, -3], [1, 1, -1, 0, 2], [-4, -2, 1, 2, 0], [3, 3, -4, 3, 8], [6, 1, 3, -9, -11] ])

Convert to a sympy Matrix for easier manipulation

matrix = Matrix(vectors)

Perform row reduction to find the basis

reduced_matrix = matrix.rref()[0]

Extract the basis vectors

basis_vectors = np.array(reduced_matrix)

import pandas as pd df = pd.DataFrame(basis_vectors) import ace_tools as tools; tools.display_dataframe_to_user(name="Basis Vectors in Row Reduced Form", dataframe=df)The basis vectors of the space are displayed in the row-reduced matrix form. If you have any further questions or need clarification, feel free to ask!

Relative Questions:

1. How do we determine linear independence in a set of vectors?
2. Why is row reduction helpful in finding the basis of a vector space?
3. Can a set of vectors in higher dimensions still form a basis for a smaller-dimensional space?
4. What properties does the row-reduced echelon form have that make it unique?
5. How do changes in the field of numbers (e.g., integers, rationals) affect the basis calculation?

Tip:

Row reduction is a powerful technique not only for finding bases but also for solving systems of linear equations efficiently.