How this calculator works
Reduce the matrix to reduced row echelon form. Columns without a pivot correspond to free variables. Setting one free variable to one and the others to zero, then solving for the pivot variables, gives one basis vector per free variable. The number of basis vectors is the nullity, and it equals the number of columns minus the rank. The null space always contains the zero vector even when its dimension is zero.
Formula
Null(A) = {x : Ax = 0}. Nullity(A) = columns(A) - rank(A).
Worked example
The matrix with rows 1, 2, 3 and 2, 4, 6 reduces to 1, 2, 3; 0, 0, 0. Columns 2 and 3 are free. Setting x2 = 1 gives -2, 1, 0, and setting x3 = 1 gives -3, 0, 1. Those two vectors form a basis and the nullity is 2.
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Frequently asked questions
Is the basis unique?
No. Any linearly independent set spanning the null space is a basis. The free-variable choice gives one convenient answer, not the only one.
When is the null space trivial?
When the columns are linearly independent, which for a square matrix means the matrix is invertible. Then the only solution of Ax = 0 is x = 0.
How is nullity related to rank?
The rank-nullity theorem says the rank plus the nullity equals the number of columns. Finding one of them therefore gives the other.