4.6: Rank

Definition:Let A be an mxn matrix. Then each row of A has n entries and can therefore be associated with a vector in

The set of all linear combinations of the row vectors in A is calledthe row space of A.

Example:

A

A

of Space Row

351371

1719113

51531

170852

n

Theorem 13

If two matrices A and B are row equivalent, then their row spacesare the same.

If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B.

rref

Example: Find the bases for the row space and column space of the matrix A.

2402

1243

1603

0110

3131

:A

0000

0000

1000

0110

3131

:)(Arref

Definition

The rank of A is the dimension of the column space of A.

Theorem 14 (The Rank Theorem)

The dimensions of the column space and the row space of an matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies

nm

nAA Nuldimrank

nAA nullity rank

Example:1. If A is a matrix with a three-dimensional null space,

what is the rank of A?

2. Could a matrix have a three-dimensional null space?

108

125

Example:A scientist has found two solutions to a homogeneous system of 40 equations in 42 variables. The two solutions are not multiples, and all other solutions can be constructed by adding together appropriate multiples of these two solutions. Can the scientist be certain that an associated nonhomogeneous system (with the same coefficients) has a solution?

The Invertible Matrix Theorem (Continued)

Let A be an matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix.

m. The columns of A form a basis of .

n.

o.

p.

q.

r.

nn

n

A Col

A Col dim

Arank

A Nul

ANullity

n

n

n

}0{

0

Definition

The rank of A is the dimension of the column space of A.

Theorem 14 (The Rank Theorem)

The dimensions of the column space and the row space of an matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies

nm

nAA Nuldimrank

nAA nullity rank