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Such spaces will be discussed in Section 2-3. When considering whether certain mathematical objects constitute a vector space, one must be careful to ensure that each of the above postulates is satisfied. It is easily verified that the geometric vectors in two- and three-dimensional space discussed previously satisfy these postulates. Let us now consider examples of other kinds of mathematical objects that also behave as vectors in a vector space, that perhaps are not so obvious. As a first example, let us consider a set of vectors that is more general than those encountered previously, in which there are more than three components for each vector.

For example, the "position" of a particle in classical mechanics at a given time is uniquely defined as being some single point in space. On the other hand, the concept of "position" when discussing wave phenomena is totally inappropriate in classical mechanics, since a wave extends throughout all space. , an electron that exhibits both wave and particle characteristics is not obvious. As we shall see in later chapters, these two concepts can be reconciled by means of the uncertainty principle, which can be thought of as a principle that "blurs" the notion of simultaneously specifying the position and momentum of a particle, while "sharpening" the wave picture somewhat by treating "wave packets," formed by suitable superpositions of waves to give reasonably localized wave packets that can be thought of as resembling particles.

U', Vj)= (u- ± CiVi, Vj) (2-138) 1=1 = (u, Vj) - n ~ ci(Vi, vJ> i=1 n = (u, Vj)- ~ i=1 CiOij (2-139) (2-140) or (u', Vj)=cj-cj*=O, (2-141) for j = 1, 2, ... , n, which completes the proof. Note also that by combining Eqs. (2-132), (2-134), and (2-136), we have n Iu' 12= lu 12_ ~ ICiI2, i=1 (2-142) which can be rearranged to give n IU 12 = ~ IC;! 2+ Iu ' 12. i=1 (2-143) We have shown above that u' is orthogonal to the space spanned by {v I> V2, ••• , Vn }. •. • , Vn } is complete. This implies that {V I> V2J •..