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The set of real numbers.

The Concept of a Vector Space

In physics one often meets physical quantities which have both a magnitude and a direction. For example, a force is described by its magnitude and the direction in which it is applied. Such quantities are often called vectors by the physicist. For simplicity, we will confine our discussion to vectors in two dimensions. A convenient representation of a vector quantity can be obtained by drawing an arrow from the origin of a rectangular coordinate system in the direction of the vector, where the length of the arrow equals the magnitude of the vector (see Figure 1). It is a short abstract jump to replace the arrow by the ordered pair (a,b) of real numbers, giving the coordinates of the endpoint of the arrow. Certainly, all the information conveyed by the arrow is provided by the ordered pair (a,b). Thus for the mathematician, the vector of the physicist (at least in two dimensions) is an ordered pair of real numbers.

vector in a plane
Figure 1: A Vector in the Plane.

Suppose that forces F1 and F2 are applied to a body. Experiment has shown the physicist that the resulting force, which is called the sum of F1 and F2, is the same as the force described by constructing a parallelogram with F1 and F2 as the sides and taking the diagonal of this parallelogram (see Figure 2). Thus if F1 and F2 correspond to the ordered pairs (a,b), (c,d) respectively, then a casual reference to the figure should convince the reader that F1 + F2 correspond to (a + c, b + d).

Addition of Vectors
Figure 2: Addition of Vectors.

The mathematician interprets addition of vectors as follows: Let V2 denote the set of all ordered pairs (a,b)(a,b element of R). Define an operation +, called addition, on V2 by setting

(a, b) + (c, d) = (a + c, b + d).

It is trivial to verify that with respect to this operation, V2 is an abelian group. The identity element is (0,0), whereas the inverse of (a,b) is (-a,-b).

Scalar Multiplication
Figure 3: Scalar Multiplication.

In addition to the operation + among vectors, there is another natural operation present. If F is a vector and alpha is a real number, then it is possible to "multiply" alpha and F as follows: alphaF is the vector whose magnitude is |alpha| · |F|, where |F| denotes the magnitude of F. The direction of alphaF is the same as the direction of F if alpha > 0, and is the opposite direction to F if alpha < 0 (see Figure 3). Thus, if F corresponds to (a,b), then alphaF corresponds to (alphaa,alphab). To distinguish the real numbers from the vectors, the real numbers are called scalars. Thus, we have defined a multiplication which allows taking the product of a scalar and a vector. This multiplication is called scalar multiplication and has the following properties: If alpha,beta element of R, v,w element of V2, then

(alpha + beta) · v = alpha · v + beta · v,
alpha · (v + w) = alpha · v + alpha · w,
(alphabeta) · v = alpha · (betav),
1 · v = v.

For example, let us prove (1): Let v = (a,b). Then

(alpha + beta) · v = (alpha + beta) · (a · b)
= ((alpha + beta)a, (alpha + beta)b)
=(alphaa + betaa, alphab + betab)
= (alphaa, alphab) + (betaa, betab)
= alpha · (a,b) + beta · (a,b)
= alpha · v + beta · v.

The proofs of (2) - (4) are similar.

We have just created a new algebraic system. The vectors in two dimensions form an abelian group with respect to addition, but they also admit multiplication by a set of elements known as scalars, such that the properties (1)-(14) are satisfied. Such an algebraic system is called a vector space. Let us now formally define this concept.

Definition 1: Let F be a field. A vector space over F is a nonempty set V plus two functions +: V cross V maps V, ·: F cross V maps V, called vector addition and scalar multiplication, respectively, such that the following properties are satisfied:

V1. With respect to vector addition, V is an abelian group.

V2. For all alpha element of F, v,w element of V, we have alpha · (v + w) = alpha · v + alpha · w.

V3. For all alpha,beta element of F, v element of V, we have (alpha + beta) · v = alpha · v + beta · v.

V4. For all alpha,beta element of F, v element of V, we have alpha · (beta · v) = (alpha · beta) · v.

V5. For all v element of V, we have 1 · v = v.

The elements of V are called vectors and the elements of F are called scalars.

Example 1: F = R, V = V2. Then V2 is a vector space over R.

Example 2: Let F be any field. V = {a1,...,an) | ai element of F, 1 < i < n}. Define vector addition in V by

(a1,...,an) + (b1,...,bn) = (a1 + b1,...,an + bn).

Define scalar multiplication by

alpha · (a1,...,an) = (alphaa1,...,alphaan)    (alpha element of F).

Then with respect to these operations, V becomes a vector space over F, denoted Fn.

Example 3: Let F be any field, V = F[X], where X is an indeterminate over F. Define vector addition to be the usual addition for polynomials and define scalar multiplication as follows: Let alpha element of F, f = a0 + a1X +...+ anXn element of F[X]. Set

alpha · f = alphaa0 + (alphaa1)X +...+ (alphaan)Xn.

Then with respect to these operations, F[X] becomes a vector space over F.

Example 4: Let F be any field, V = {f element of F[X] | deg(f ) < n}, and let vector addition and scalar multiplication be defined as in example 3. Then V is a vector space over F.

Example 5: Let C[0,1] denote the set of all real valued functions which are continuous on the closed interval [0,1]. Let F = R, V = C[0,1]. Define vector addition by

(f + g)(x) = f(x) + g(x)   (x element of [0,1]),

and define scalar multiplication by

(alphaf)(x) = alphaf(x)   (x element of [0,1]).

Then C[0,1] is a vector space over R.

Example 6: Let E be a field, F a subfield of E. Let V = E and define vector addition to be the addition in E. Let scalar multiplication be the multiplication in E. Then E is a vector space over F.

Thus, we see from the above examples that a great variety of mathematical objects qualify as vector spaces. The notion of a vector space is one of the fundamental unifying notions in modern mathematics, both pure and applied, and finds applications in fields as diverse as psychology and quantum mechanics.

Some Remarks: 1. If V is a vector space with vector addition +, then V is an abelian group with respect to +. In order to minimize confusion, we will always use additive notation with respect to this group. The identity element will be denoted 0, whereas the inverse of v will be denoted -v. Note that 0 is not the same as 0 element of F.

2. Vectors will always be denoted by lower case Roman boldface letters, for example v,w,x,y. Scalars will be denoted by lower case Greek letters, for example alpha,beta,gamma,delta.

3. We will usually omit the · signifying scalar multiplication and will write alphav instead of alpha · v.

4. The reason for the axiom 1 · v = v(v element of V) in our definition of a vector space is to eliminate degenerate examples for which alphav = 0 for all v element of V, which satisfy the remaining axioms.

Three elementary, but useful, facts concerning vector spaces are

0v = 0    (v element of V),
(-1)v = -v    (v element of V),
alpha(-v) = (-alpha)v = -(alphav)  (alpha element of F, v element of V),