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.

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).

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

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).

In addition to the operation + among vectors, there is another natural operation present. If F is a vector and is a real number, then it is possible to "multiply" and F as follows: F is the vector whose magnitude is || · |F|, where |F| denotes the magnitude of F. The direction of F is the same as the direction of F if > 0, and is the opposite direction to F if < 0 (see Figure 3). Thus, if F corresponds to (a,b), then F corresponds to (a,b). 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 , R, v,w V₂, then

( + ) · v = · v + · v,

· (v + w) = · v + · w,

() · v = · (v),

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

( + ) · v = ( + ) · (a · b)

= (( + )a, ( + )b)

=(a + a, b + b)

= (a, b) + (a, b)

= · (a,b) + · (a,b)

= · v + · 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 V V, ·: F V 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 F, v,w V, we have · (v + w) = · v + · w.

V₃. For all , F, v V, we have ( + ) · v = · v + · v.

V₄. For all , F, v V, we have · ( · v) = ( · ) · v.

V₅. For all v 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,...,a_n) | a_i F, 1 < i < n}. Define vector addition in V by

Define scalar multiplication by

· (a₁,...,a_n) = (a₁,...,a_n)    ( 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 F, f = a₀ + a₁X +...+ a_nXⁿ F[X]. Set

· f = a₀ + (a₁)X +...+ (a_n)Xⁿ.

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

Example 4: Let F be any field, V = {f 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 [0,1]),

and define scalar multiplication by

(f)(x) = f(x)   (x [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 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 ,,,.

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

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

Three elementary, but useful, facts concerning vector spaces are

0v = 0    (v V),

(-1)v = -v    (v V),

(-v) = (-)v = -(v)  ( F, v V),