### Rank and Nullity

##### 2. Properties of rank and nullity

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

rankA = rankAT.

Example From the computation

 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 R6) 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, rankAm and n. Moreover,

• rankA = m ⇔ Columns of A span RmAx = b has solutions for all b.
• rankA = n ⇔ Columns of A are linearly independent ⇔ Solution of Ax = b is unique.

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.

• A is invertible.
• colA = Rm.
• rankA = n.
• nulA = {0}.
• nullityA = 0.