Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
We now proceed to generalize the concept of a vector space. An arbitrary set of
n objects j1i; j2i; j3i; ...; ji; ...; j'i form a linear vector V n if these objects, called
vectors, meet the following axioms or properties:
A.1 If ji and j'
i are objects in V n and k is a scalar, then ji‡ j'
i and kji are
in V n , a feature called closure.
A.2 ji‡ j'
i ˆ j'
i‡ ji; that is, addition is commutative.
A.3 ( ji‡ j'
i†‡ j i ˆ ji‡…'
j i‡ j i); that is, addition is associative.
A.4 k… ji‡ j'
i† ˆ kji‡ k j'
i; that is, scalar multiplication is distributive in the
vectors.
A.5 …k ‡ † ji ˆ kji‡ j i; that is, scalar multiplication is distributive in the
scalars.
A.6 k…ji† ˆ kji; that is, scalar multiplication is associative.
A.7 There exists a null vector j0i in V n such that ji‡ jiˆ 0 ji for all ji in V n .
A.8 For every vector ji in V n , there exists an inverse under addition, ji such
that ji‡
j i ˆ j0i.
The set of numbers a; b; ...used in scalar multiplication of vectors is called the
®eld over which the vector ®eld is de®ned. If the ®eld consists of real numbers, we
have a real vector ®eld; if they are complex, we have a complex ®eld. Note that the
vectors themselves are neither real nor complex, the nature of the vectors is not
speci®ed. Vectors can be any kinds of objects; all that is required is that the vector
space axioms be satis®ed. Thus we purposely do not use the symbol V to denote
the vectors as the ®rst step to turn the reader away from the limited concept of the
vector as a directed line segment. Instead, we use Dirac's ket and bra symbols, j i
and h j, to denote generic vectors.
The familiar three-dimensional space of position vectors E 3 is an example of
a vector space over the ®eld of real numbers. Let us now examine two simple
examples.
Example 5.1
Let V be any plane through the origin in E 3 . We wish to show that the points in
the plane V form a vector space under the addition and scalar multiplication
operations for vector in E 3 .
Solution: Since E 3 itself is a vector space under the addition and scalar multiplication
operations, thus Axioms A.2, A.3, A.4, A.5, and A.6 hold for all points
in E 3 and consequently for all points in the plane V. We therefore need only show
that Axioms A.1, A.7, and A.8 are satis®ed.
Now the plane V, passing through the origin, has an equation of the form
ax 1 ‡ bx 2 ‡ cx 3 ˆ 0:
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