Null Space

The null space $N(A)$ is the set of all solutions to $Ax = 0$:

$$N(A) = \{x : Ax = 0\}$$

Key Facts

• $N(A)$ is a subspace of $\mathbb{R}^n$ (for $m \times n$ matrix $A$)
• The dimension of $N(A) = n - \text{rank}(A)$ (the number of free variables)

How to Find It

1. Row reduce $A$ to echelon form
2. Identify free variables (columns without pivots)
3. Set each free variable to 1 (one at a time, rest to 0)
4. Solve for the pivot variables

Complete Solution

For $Ax = b$:

$$x = x_p + x_n$$

Where $x_p$ is a particular solution and $x_n$ is any vector in $N(A)$.