The null space (or kernel) of a matrix \(A\) is the set of all vectors \(x\) that satisfy the homogeneous equation \(Ax = 0\).

The Process:

• 1. Augment the Matrix: Create an augmented matrix by adding a column of zeros to the right of matrix A, forming \([A|0]\).
• 2. Row Reduce: Use Gauss-Jordan elimination to transform the augmented matrix into its Reduced Row Echelon Form (RREF).
• 3. Find the Solution: Express the pivot variables (those in columns with a leading 1) in terms of the free variables (those in columns without a leading 1).
• 4. Form the Basis: The vectors associated with the free variables in the parametric vector form of the solution form a basis for the null space.

If there are no free variables, the only solution is the trivial one (\(x=0\)), and the null space consists only of the zero vector.