## 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

## Leave a Reply

You must be logged in to post a comment.