Numerical Methods and Optimization

18 Numerical Methods and Optimization: An Introduction

1.4.7 Rank of a matrix

Definition 1.28 (Row (Column) Space) Given an m ×n matrix A,

its row (column) space is the subspace of IR

n

(IR

m

) spanned by the row

(column) vectors of A.

The column space of A, which is given by {y ∈ IR

m

: y = Ax, x ∈ IR

n

},

is also called the range space of A.Theset{x ∈ IR

n

: Ax =0} is called the

kernel or null space of A and is denoted by Ker(A) or Null(A).

Theorem 1.8 (Rank of a matrix) The dimension of the row space of

any matrix A is equal to the dimension of its column space and is called

the rank of A (rank(A)).

Example 1.12 The rank of

A =

⎡

⎣

123

456

579

⎤

⎦

is 2, since the first ...