Page 425 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Thus there are four parameters.

Suppose now that A is an matrix of rank r; it follows from Theorem 5.6.2 that is an matrix of rank r. Applying

Theorem 5.6.3 to A and yields

from which we deduce the following table relating the dimensions of the four fundamental spaces of an matrix A of rank
r.

Fundamental Space Dimension

Row space of A r
Column space of A r
Nullspace of A
Nullspace of

Applications of Rank

The advent of the Internet has stimulated research on finding efficient methods for transmitting large amounts of digital

data over communications lines with limited bandwidth. Digital data is commonly stored in matrix form, and many

techniques for improving transmission speed use the rank of a matrix in some way. Rank plays a role because it measures

the “redundancy” in a matrix in the sense that if A is an matrix of rank k, then of the column vectors and

of the row vectors can be expressed in terms of k linarly independent column or row vectors. The essential idea in many

data compression schemes is to approximate the original data set by a data set with smaller rank that conveys nearly the

same information, then eliminate redundant vectors in the approximating set to speed up the transmission time.

Maximum Value for Rank

If A is an matrix, then the row vectors lie in and the column vectors lie in . This implies that the row space of A is

at most n-dimensional and that the column space is at most m-dimensional. Since the row and column spaces have the same

dimension (the rank of A), we must conclude that if , then the rank of A is at most the smaller of the values of m and n.

We denote this by writing

(5)

where denotes the smaller of the numbers m and n if or denotes their common value if .

EXAMPLE 4 Maximum Value of Rank for a Matrix

If A is a matrix, then the rank of A is at most 4, and consequently, the seven row vectors must be linearly dependent. If A
is a matrix, then again the rank of A is at most 4, and consequently, the seven column vectors must be linearly dependent.