columns of any upper as well as lower triangular matrix will fill that space or linearly independent

Why gaussian elimination also gives information regarding linear independence of the columns as well as rows?

Elimination with matrices

multiplication and inverses

Factorizing A into LU

Transposes Permuations and Spaces R^n

column space and null space

Solving Ax=0

Solving Ax=b

Independence, Basis and Dimension

The Four Fundamental Subspaces

Matrix Spaces Small World Graphs

Graphs Networks Incidence matrices Kirchhoff’s Laws

Orthogonal Vectors and Subspaces

Projections onto Subspaces

Projection Matrices and Least Squares

Orthogonal Matrices and Gram Schmidt

Properties of Determinants

Determinant formulas and Cofactors

Cramer’s Rule, Inverse Matrix, and Volume