A First Course in Linear Algebra, 2017a

4.10. Spanning, Linear Independence and Basis in R n 215
of the null space. Recall also that the number of leading ones in the reduced row-echelon form equals the
number of pivot columns, which is the rank of the matrix, which is the same as the dimension of either the
column or row space.
Before we proceed to an important theorem, we first define what is meant by the nullity of a matrix.
Definition 4.111: Nullity
The dimension of the null space of a matrix is called the nullity, denoted dim(null(A)).
From our observation above we can now state an important theorem.
Theorem 4.112: Rank and Nullity
Let A be an m × n matrix. Then rank(A)+dim(null(A)) = n.
Consider the following example, which we first explored above in Example 4.109
Example 4.113: Rank and Nullity
Let
Find rank(A) and dim(null(A)).
⎡
A = ⎣
1 2 1
0 −1 1
2 3 3
⎤
⎦
Solution. In the above Example 4.109 we determined that the reduced row-echelon form of A is given by
⎡
1 0
⎤
3
⎣ 0 1 −1 ⎦
0 0 0
Therefore the rank of A is 2. We also determined that the null space of A is given by
⎧⎡
⎤⎫
⎨ −3 ⎬
null(A)=span ⎣ 1 ⎦
⎩ ⎭
1
Therefore the nullity of A is 1. It follows from Theorem 4.112 that rank(A)+dim(null(A)) = 2+1 = 3,
which is the number of columns of A.
♠
We conclude this section with two similar, and important, theorems.