Rank of a matrix and the rank-nullity theorem
If you find proofs shorter than mine, shoot me an email.
Rank and its properties
Definition 1. The dimension of the image is called a rank of the matrix and denoted .
Exercise 1. For any matrix
Proof. Suppose is of size . We know that
By (2) any can be represented as
(3) with and
From (3) By Definition 1, is spanned by some linearly independent vectors with Hence with some depending on and This shows that is spanned by If we prove linear independence of these vectors, (1) will follow.
Suppose with some Then This tells us that Since the vector also belongs to by Exercise 3 it is zero. By linear independence, is possible only when This proves the linear independence of and the statement.
Exercise 2 (rank-nullity theorem) If is then
Proof. Exercise 4 and equation (2) imply But we know from Exercise 1 that
Exercise 3. Let be of size Then
Proof. From Exercise 2 we see that Applying this inequality to we get