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
(1)
Proof. Suppose is of size
. We know that
(2)
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
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