# Kernel of a linear function: Nullspace

### 5.47.13 Kernel of a linear function: Nullspace

The Nullspace command is the inert form of nullspace. Warning:
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 Apply).

- Nullspace takes one argument:
  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.

- Input:
  Nullspace([[1,1,2],[2,1,3],[3,1,4]])
  Output:

  nullspace

  ⎛
  ⎜
  ⎜
  ⎝

  ⎡
  ⎢
  ⎢
  ⎣

  1 | 1 | 2
  2 | 1 | 3
  3 | 1 | 4

  ⎤
  ⎥
  ⎥
  ⎦

  ⎞
  ⎟
  ⎟
  ⎠

- Input (in Maple mode):
  Nullspace([[1,2],[3,1]]) mod 5
  Output:

  ⎡ ⎣ | 2,−1 | ⎤ ⎦

  In Xcas mode, the equivalent input is:
  Input (in Xcas mode):
  nullspace([[1,2],[3,1]] % 5)
  Output:

  ⎡
  ⎣

  2%5 | −1

  ⎤
  ⎦