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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