MATLAB Function Reference |

**Syntax**

**Description**

```
Z = null(A)
```

is an orthonormal basis for the null space of `A`

obtained from the singular value decomposition. That is, `A*Z`

has negligible elements, `size(Z,2)`

is the nullity of `A`

, and `Z'*Z = I`

.

```
Z = null(A,'r')
```

is a "rational" basis for the null space obtained from the reduced row echelon form. `A*Z`

is zero, `size(Z,2)`

is an estimate for the nullity of `A`

, and, if `A`

is a small matrix with integer elements, the elements of the reduced row echelon form (as computed using `rref`

) are ratios of small integers.

The orthonormal basis is preferable numerically, while the rational basis may be preferable pedagogically.

**Example**

**Example 1.** Compute the orthonormal basis for the null space of a matrix A.

A = [1 2 3 1 2 3 1 2 3]; Z = null(A) Z = 0.9636 0 -0.1482 -0.8321 -0.2224 0.5547 A*Z ans = 1.0e-015 * 0.2220 0.2220 0.2220 0.2220 0.2220 0.2220 Z'*Z ans = 1.0000 -0.0000 -0.0000 1.0000

**Example 2.** Compute the rational basis for the null space of the same matrix `A`

.

**See Also**

now | num2cell |