How to Calculate Null Space

What is Null Space?

The null space (kernel) of a matrix A is the set of all vectors x such that Ax = 0. The nullity is the dimension of the null space. Understanding null space is essential in solving linear systems and linear algebra.

Formula

Null space of A: all vectors x where Ax = 0

A: matrix
x: vector in null space
Null(A): null space (kernel) of A

Step-by-Step Guide

1Solve Ax = 0 using row reduction (RREF)
2Free variables correspond to null space dimensions
3Rank-nullity theorem: rank + nullity = n (columns)
4Null space is always a subspace containing the zero vector

Worked Examples

Input

[[1,2,3],[4,5,6]] × x = 0

Result

Null space has dimension 1; one free variable

Frequently Asked Questions

Is the zero vector always in the null space?

Yes, A × 0 = 0 always. So the null space is never empty.

What is the relationship between null space and rank?

Rank-nullity theorem: rank(A) + nullity(A) = number of columns. (Nullity = dimension of null space.)

Can the null space be trivial (only zero)?

Yes, if A has full column rank. If A has dependent columns, null space is larger.

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