Linear Algebra: Basis Reduction and Extension Theorems

Published on October 31, 2025

Linear-Algebra

This problem proves that any spanning set can be reduced to a basis and any linearly independent set can be extended to a basis in a vector space.

Question

solve this Linear Algebra problem

The eigenvalues of the given 3x3 matrix are 1, 2, and 2.

The determinant of the given 4x4 matrix is calculated using cofactor expansion along the fourth row, resulting in a determinant of -21.

The determinant of the 2x2 matrix [[2, 2], [3, 5]] is calculated as (2 * 5) - (2 * 3) = 4.

The determinant of the 3x3 matrix is calculated using the formula a(ei - fh) - b(di - fg) + c(dh - eg), resulting in a determinant of 0.