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Linear operators

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Linear operators
8.2 LINEAR OPERATORS
where the equality holds if the sum includes all N basis vectors. If not
all the basis vectors are included in the sum then the inequality results
(though of course the equality remains if those basis vectors omitted all
have ai = 0). Bessel’s inequality can also be written
a|a ≥
|ai |2 ,
i
where the ai are the components of a in the orthonormal basis. From (8.16)
these are given by ai = êi |a. The above may be proved by considering
2 &
'
a −
= a −
ê
|aê
ê
|aê
êj |aêj .
a −
i
i
i
i
i
i
j
Expanding out the inner product and using êi |a∗ = a|êi , we obtain
2
a −
êi |aêi = a|a − 2
a|êi êi |a +
a|êi êj |aêi |êj .
i
i
i
j
Now êi |êj = δij , since the basis is orthonormal, and so we find
2
êi |aêi = a2 −
|êi |a|2 ,
0 ≤ a −
i
i
which is Bessel’s inequality.
We take this opportunity to mention also
(iv) the parallelogram equality
a + b2 + a − b2 = 2 a2 + b2 ,
(8.22)
which may be proved straightforwardly from the properties of the inner
product.
8.2 Linear operators
We now discuss the action of linear operators on vectors in a vector space. A
linear operator A associates with every vector x another vector
y = A x,
in such a way that, for two vectors a and b,
A (λa + µb) = λA a + µA b,
where λ, µ are scalars. We say that A ‘operates’ on x to give the vector y. We
note that the action of A is independent of any basis or coordinate system and
247
MATRICES AND VECTOR SPACES
may be thought of as ‘transforming’ one geometrical entity (i.e. a vector) into
another.
If we now introduce a basis ei , i = 1, 2, . . . , N, into our vector space then the
action of A on each of the basis vectors is to produce a linear combination of
the latter; this may be written as
N
A ej =
Aij ei ,
(8.23)
i=1
where Aij is the ith component of the vector A ej in this basis; collectively the
numbers Aij are called the components of the linear operator in the ei -basis. In
this basis we can express the relation y = A x in component form as
y=
N
i=1


N
N
N
yi ei = A 
xj ej  =
xj
Aij ei ,
j=1
j=1
i=1
and hence, in purely component form, in this basis we have
yi =
N
Aij xj .
(8.24)
j=1
If we had chosen a different basis ei , in which the components of x, y and A
are xi , yi and Aij respectively then the geometrical relationship y = A x would be
represented in this new basis by
yi =
N
Aij xj .
j=1
We have so far assumed that the vector y is in the same vector space as
x. If, however, y belongs to a different vector space, which may in general be
M-dimensional (M = N) then the above analysis needs a slight modification. By
introducing a basis set fi , i = 1, 2, . . . , M, into the vector space to which y belongs
we may generalise (8.23) as
A ej =
M
Aij fi ,
i=1
where the components Aij of the linear operator A relate to both of the bases ej
and fi .
248
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