A First Course in Linear Algebra, 2017a

4.10. Spanning, Linear Independence and Basis in R n 211
Corollary 4.103: Results of the Rank Theorem
Let A be a matrix. Then the following are true:
1. rank(A)=rank(A T ).
2. For A of size m × n, rank(A) ≤ m and rank(A) ≤ n.
3. For A of size n × n, A is invertible if and only if rank(A)=n.
4. For invertible matrices B and C of appropriate size, rank(A)=rank(BA)=rank(AC).
Consider the following example.
Example 4.104: Rank of the Transpose
Let
Find rank(A) and rank(A T ).
[
A =
1 2
−1 1
]
Solution. To find rank(A) we first row reduce to find the reduced row-echelon form.
[ ] [ ]
1 2
1 0
A = →···→
−1 1
0 1
Therefore the rank of A is 2. Now consider A T given by
[ ]
A T 1 −1
=
2 1
Again we row reduce to find the reduced row-echelon form.
[ ] [ 1 −1
1 0
→···→
2 1
0 1
]
You can see that rank(A T )=2, the same as rank(A).
♠
We now define what is meant by the null space of a general m × n matrix.
Definition 4.105: Null Space, or Kernel, of A
The null space of a matrix A, also referred to as the kernel of A, is defined as follows.
{ }
null(A)= ⃗x : A⃗x =⃗0
It can also be referred to using the notation ker(A). Similarly, we can discuss the image of A, denoted
by im(A). TheimageofA consists of the vectors of R m which “get hit” by A. The formal definition is as
follows.