Mathematical Methods for Physicists: A concise introduction - Site Map

GENERAL LINEAR VECTOR SPACES
We usually de®ne the inner product of two vectors in E 3 in terms of lengths of
the vectors and the angle between the vectors: A B ˆ AB cos ; ˆ€…A; B†. We
do not de®ne the inner product in E n in the same manner. However, the inner
product in E 3 has a second equivalent expression in terms of components:
A B ˆ A 1 B 1 ‡ A 2 B 2 ‡ A 3 B 3 . We choose to de®ne a similar formula for the general
case. We made this choice because of the further generalization that will be
outlined in the next section. Thus, for any two vectors u ˆ…u 1 ; u 2 ; ...; u n † and
v ˆ…v 1 ; v 2 ; ...; v n † in E n , the inner (or dot) product u v is de®ned by
u v ˆ u 1
*v 1 ‡ u 2
*v 2 ‡‡u n
*v n
…5:6†
where the asterisk denotes complex conjugation. u is often called the prefactor
and v the post-factor. The inner product is linear with respect to the post-factor,
and anti-linear with respect to the prefactor:
u …av ‡ bw† ˆau v ‡ bu w;
…au ‡ bv†w ˆ a*…u v†‡b*…u w†:
We expect the inner product for the general case also to have the following three
main features:
u v ˆ…v u†*
u …av ‡ bw† ˆau v ‡ bu w
u u 0 …ˆ 0; if and only if u ˆ 0†:
…5:7a†
…5:7b†
…5:7c†
Many of the familiar ideas from E 2 and E 3 have been carried over, so it is
common to refer to E n with the operations of addition, scalar multiplication, and
with the inner product that we have de®ned here as Euclidean n-space.
General linear vector spaces
We now generalize the concept of vector space still further: a set of `objects' (or
elements) obeying a set of axioms, which will be chosen by abstracting the most
important properties of vectors in E n , forms a linear vector space V n with the
objects called vectors. Before introducing the requisite axioms, we ®rst adapt a
notation for our general vectors: general vectors are designated by the symbol ji,
which we call, following Dirac, ket vectors; the conjugates of ket vectors are
denoted by the symbol hj, the bra vectors. However, for simplicity, we shall
refer in the future to the ket vectors ji simply as vectors, and to the hjsas
conjugate vectors. We now proceed to de®ne two basic operations on these
vectors: addition and multiplication by scalars.
By addition we mean a rule for forming the sum, denoted j 1 i‡j 2 i, for
any pair of vectors j 1 i and j 2 i.
By scalar multiplication we mean a rule for associating with each scalar k
and each vector j i a new vector kj i.
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