Mathematical Physics Vol 1

Chapter 1. Vector algebra

34

In a normed vector space the metric is introduced by

d ( x , y )= ∥ x − y ∥ .

(1.63)

Let us denote by R n a set whose points are ordered n –tuples of real (complex) numbers and introduce a metric in this space by d ( x , y )= s n ∑ i = 1 ( x i − y i ) 2 . (1.64) The space R n is called Euclidian (real or complex) metric space, and d defined in this way satisfies the conditions (1.59)–(1.62). Let us now introduce the concept of linear operator. Consider a linear vector function, of a vector variable, which assigns to each vector x another vector A ( x ) , and for which the following is true A ( α x + β y )= α A ( x )+ β A ( y ) , (1.65) where α and β are scalars, and x and y vectors. The function defined in this way is called linear operator . A linear operator is fully determined if vectors A ( e i ) are given, where vectors e i ( i = 1 , 2 ,..., n ) form a set of base vectors. Vectors A ( e i ) can be expressed using the vectors e i

n ∑ j = 1

A ( e i )=

A ji e j ,

(1.66)

where A ji is the j –th component of the vector A ( e i ) . Let us now consider an arbitrary vector x , and denote A ( x ) by y , i.e. A ( x )= y . Then the following relations can be established using (1.65) and (1.66). Let us first express y using base vectors

n ∑ j = 1

y =

y j e j .

Given that

x i e i ! =

n ∑ i = 1

y = A ( x )= A

(1.67)

A ji x i ! e j ,

n ∑ i = 1

n ∑ i = 1

n ∑ j = 1

n ∑ j = 1

n ∑ i = 1

x i A ( e i )=

A ji e j =

x i

=

it follows that vector coordinates x and y are bound by the relation

n ∑ i = 1

y j =

A ji x i .

(1.68)

This can also be expressed in the following way. Let us assume that we have established the relation between vectors x and y using a linear operator A applied to x . This can be denoted symbolically by 17 y = A x . (1.69) 17 Note that in the case of linear operator A , the symbol A x is used on an equal basis.