The first property is that the rank is not changed by the transpose. The proof can be found here.

rank[ |
1 | 3 | 2 | 0 | 1 | 0 | ] = 4 |

-1 | -1 | -1 | 1 | 0 | 1 | ||

0 | 4 | 2 | 4 | 3 | 3 | ||

1 | 3 | 2 | -2 | 0 | 0 |

in an earlier example, we have

rank[ |
1 | -1 | 0 | 1 | ] = 4, nullity[ |
1 | -1 | 0 | 1 | ] = 4 - 4 = 0. |

3 | -1 | 4 | 3 | 3 | -1 | 4 | 3 | |||

2 | -1 | 2 | 2 | 2 | -1 | 2 | 2 | |||

0 | 1 | 4 | -2 | 0 | 1 | 4 | -2 | |||

1 | 0 | 3 | 0 | 1 | 0 | 3 | 0 | |||

0 | 1 | 3 | 0 | 0 | 1 | 3 | 0 |

Note that since the nullity of the 6 by 4 matrix is zero, the null space consists of only the zero vector. This means that the four column vectors (of **R**^{6}) are linearly independent.

Since the rank means the "essential size", it is always no bigger than the "apparent size".

For an `m` by `n` matrix ** A**,

`rank`=**A**`m`⇔ Columns ofspan`A`**R**^{m}⇔=`Ax`has solutions for all`b`.`b``rank`=**A**`n`⇔ Columns ofare linearly independent ⇔ Solution of`A`=`Ax`is unique.`b`

The proof can be found here. As a consequence of the property, we may characterize the invertibility of a square matrix by the rank and nullity (see this version of the basic principal of linear algebra).

For an `n` by `n` matrix ** A**, the following are equivalent.

is invertible.`A``col`=**A****R**^{m}.`rank`=**A**`n`.`nul`= {**A****0**}.`nullity`= 0.**A**

[Extra: Properties of rank and nullity]