3. The rank of a matrix, determinant

3.1. Definitions and terminology

The image R(A) of a linear mapping(a matrix) A : V -> Z is

R(A) = { | : = A}

and the nullspace (or the kernel) N(A) is

N(A) = { | A = }.

Clearly, R(A) is a subspace of Z and N(A) is a subspace of V.

The rank of A is dim R(A).

The nullity is dim N(A).

Example 1: Image, rank, nullspace and nullity

Proposition 1 The rank of a matrix A is the number of

1) linearly independent columns of A.

2) linearly independent rows of A.

Proof. 1): Let A be an m x n matrix and denote

A = (_{1 2} ..._n )

Then R(A)

<=> =		x1		: = A
		:
		xn

<=> =			x1		: = (1 ... n)		x1		=	xii
			:				:
			xn				xn

<=> L{₁ , ..., _n }

so that R(A) = L{₁ , ..., _n }

   =>

the rank of A	= dim R(A)
	= dim L{1 , ..., n }
	= the number of linearly independent vectors in {1 , ..., n }
	= the number of linearly independent columns of A

2) Omitted.

The proof of the following result, called the rank theorem, is also omitted:

Proposition 2 If A is an m x n matrix, then

For a linear mapping A : V -> Z, where V = n and dim Z = m, we have

It follows from the previous proposition, that if A : V -> Z, dim V = n, then dim R(A) = n - dim N(A) => A cannot increase the dimension of V.

Proposition 3. If A and B are matrices, then

the rank of AB min{the rank of A, the rank of B}

The elementary row operations do not affect the rank of a matrix A.