Bases for Null and Column Spaces
Reduced row echelon form reveals explicit bases for both the null space and column space of any matrix.
The Column Space Basis
The column space of a matrix consists of all possible linear combinations of its columns. To find a basis, convert the matrix to row echelon form and identify the pivot columns. The original columns corresponding to these pivot positions form a linearly independent set that spans the column space.
\text{col}(A) = \text{span}(\mathbf{a}_{j_1}, \mathbf{a}_{j_2}, \dots, \mathbf{a}_{j_r})The Null Space Basis
The null space contains all solutions to the homogeneous equation Ax = 0. By solving the system using reduced row echelon form, you can express pivot variables in terms of free variables. Factoring out the free variables produces vectors that form a basis for the null space.
\text{null}(A) = \{ \mathbf{x} \mid A\mathbf{x} = \mathbf{0} \}The Rank-Nullity Theorem
The number of basis vectors in the column space is the rank of the matrix, while the number of basis vectors in the null space is the nullity. The sum of the rank and the nullity always equals the total number of columns in the matrix. This fundamental relationship ties together pivot and free variables.
\text{rank}(A) + \text{nullity}(A) = nFlashcard
The dimension of the column space of A
Check your understanding
A 4 by 7 matrix A has 3 pivot columns in its reduced row echelon form. What is the dimension of the null space of A?