Page 14 - Vector Analysis

10 CHAPTER 1. Linear Algebra

Remark 1.33. The matrix product AB is only defined if the number of columns of A is
the same as the number of rows of B. Therefore, even if AB is defined, BA might not make
sense. When A and B are both n ˆ n square matrix, AB and BA are both defined; however,
in general AB ‰ BA.

Remark 1.34. Let v P Fn be a vector such that the k-th component of v is the same as

the (i, k)-th entry of A P M(m, n; F), and w P Fn be a vector such that the k-th component

of w is the same as the (k, j)-th entry of B P M(n, ℓ; F). Then the (i, j)-th entry of AB is

simply the inner product of v and w in Fn.

[ 1 0 ]  ´1 
0 ´1 2 2 0 1
Example 1.35. Let A = 1 and B = 1 2. Then
3
0
´1

[]
0 1 1
AB = ´4 1 ´2

but BA is not defined.

Proposition 1.36. Let A P M(m, n; F), B P M(n, ℓ; F) and C P M(ℓ, k; F). Then

A(BC) = (AB)C .

Definition 1.37 (The range and the null space of matrices). Let A P M(m, n; F). The
range of A, denoted by R(A), is the subset of Fm given by

R(A) = ␣Ax P Fm ˇ x P Fn( ,
ˇ

and the null space of A, denoted by null(A), is the subset of Fn given by

null(A) = ␣x P Fn ˇ Ax = 0( .
ˇ

Proposition 1.38. Let A P M(m, n; F). Then R(A) and null(A) are vector subspaces of Fn
and Fm, respectively.

Definition 1.39 (Kronecker’s delta). The Kronecker delta is a function, denoted by δ, of

two variables (usually positive integers) such that the function is 1 if the two variables are

equal, and 0 otherwise. When the two variables are i and j, the value δ(i, j) is usually

written as δij; that is,

" 0 if i ‰ j ,
1 if i = j .
δij =