### 15.1.11 Kernel of a linear function

The ker or kernel or nullspace command finds a basis for the kernel of a linear function.

- ker takes A, a matrix representing a linear function with respect to the standard basis.
- ker(A) returns a list of vectors that is a basis of the kernel of the linear function.
- The Nullspace command is the inert form of nullspace.

#### Example

ker([[1,1,2],[2,1,3],[3,1,4]])

⎡ ⎣ |
1 | 1 | −1
| ⎤ ⎦

##### Remark.

The Nullspace command is only useful in Maple mode. (See Section 2.5.2; you can get into Maple mode by hitting the state line red button then Prog style, then choosing Maple and clicking Apply.)

- Nullspace takes A, an integer matrix representing a linear function with respect to the standard basis.
- Nullspace(A) mod p returns a list of vectors that is a basis for the kernel of the linear transformation ℤ/pℤ[X].

#### Examples

Nullspace([[1,1,2],[2,1,3],[3,1,4]])

nullspace | ⎛ ⎜ ⎜ ⎝ |
⎡ ⎢ ⎢ ⎣ |
1 | 1 | 2
2 | 1 | 3
3 | 1 | 4
| ⎤ ⎥ ⎥ ⎦
| ⎞ ⎟ ⎟ ⎠

Input in Maple mode:

Nullspace([[1,2],[3,1]]) mod 5

| ⎡ ⎣ | 2,−1 | ⎤ ⎦

In Xcas mode, the equivalent input is:

nullspace([[1,2],[3,1]] % 5)

⎡ ⎣ |
2%5 | −1
| ⎤ ⎦