1. What is the Null Space of a Matrix?

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It represents the solutions to the homogeneous system of linear equations defined by the matrix.

2. How is the Null Space Basis Calculated?

The basis for the null space is found by solving the homogeneous system:

This is typically done using Gaussian elimination to put the matrix in reduced row echelon form, then identifying the free variables and constructing the basis vectors from the special solutions.

3. Importance of Null Space Basis

Details: The null space basis provides fundamental information about the linear transformation represented by the matrix, including its rank, dimension, and the solutions to homogeneous systems.

4. Using the Calculator

Tips: Enter the matrix with rows separated by newlines and elements within each row separated by spaces. The calculator will compute a basis for the null space of the given matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the dimension of the null space?
A: The dimension of the null space is called the nullity, which equals the number of free variables in the system Ax = 0.

Q2: How does null space relate to column space?
A: The rank-nullity theorem states that rank(A) + nullity(A) = number of columns of A.

Q3: Can the null space be empty?
A: No, the zero vector is always in the null space. The null space is empty only if it contains only the zero vector.

Q4: What if my matrix is not square?
A: The null space concept applies to any m × n matrix, not just square matrices.

Q5: How are basis vectors normalized?
A: Basis vectors are typically chosen to be linearly independent and often normalized, but normalization is not required for a basis.