Null Space Calculator
Find the null space of a 2x2 or 3x3 matrix. The null space is the set of all vectors x that satisfy Ax = 0. This calculator shows the basis vectors, nullity, rank, and determinant.
The null space (also called the kernel) of a matrix A is the set of all vectors x such that Ax = 0. It captures everything that the transformation "collapses" to zero.
Key Definitions:
- Null space: Null(A) = {x : Ax = 0}
- Nullity: The dimension of the null space (number of free variables)
- Rank: The number of pivot columns in row echelon form
- Rank-Nullity Theorem: rank + nullity = number of columns
How to Find the Null Space:
- Write out the augmented matrix [A | 0]
- Row reduce to reduced row echelon form (RREF)
- Identify pivot columns and free columns
- Express pivot variables in terms of free variables
- Write the general solution as a linear combination of basis vectors
Example:
For A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], the determinant is 0 (rank 2), so the null space is one-dimensional. Row reducing gives one free variable and the null space basis vector (1, -2, 1).
Why It Matters:
The null space tells you about the solutions to Ax = b. If the null space is trivial ({0}), the system has at most one solution. If the null space is nontrivial, there are infinitely many solutions (you can add any null space vector to a particular solution).