We collect in this section some further examples of vector spaces that indicate only a small portion of the vector spaces encountered in prac- tice.
EXAMPLE 1: (Cas a real vector space.
Recall that (C denotes the complex numbers. A complex number1
looks like
e=a+bi,
wherea, b are real numbers, andi is a number such thati2=-1. The numberais called the real part of e and the numberb the imaginary part of e. Addition of complex numbers is componentwise, i.e.,
e' +e" =(a' +a")+(b' +b")i
for complex numbers e' =a' +b'i and e" =a" +b"i. Multiplication is determined by the rule
e'e" =(a' a" - b' b")+Ca'b"+a" b')i.
The vectors in our vector space will be complex numbers. Addition of vectors is to be the ordinary addition of complex numbers, and scalar multiplication the familiar process of multiplying a complex number by a real number. With these interpretations of the basic terms,
vector ~complex number
vector addition ~addition of complex numbers
scalar multiplication ~multiplication of a complex number by a real number,
we obtain an example of a vector space. The verifications are again routine.
1We are not concerned here with the logical construction of the complex num- bers, but take their basic arithmetic properties as known. For a construction of complex numbers using ideas developed later in this book see Appendix B, so you may want to come back to this example after further study of this book.
28 3. Examples of Vector Spaces
Note that in this example we are not using all of the structure that we have, for, we may in effect, multiply two complex numbers, that is, in this example we may multiply two vectors, something it is not always possible to do in a vector space. This is a possibility2 worthy of further study, and we will do just that when we study spaces of linear transformations and linear algebras.
REMARK: The product of two polynomials of degree at mostn will have degree at most2n, but in general will not be of degree n,so you cannot multiply elements ofPnem.)in any obvious way.
Examples 2 and 3 in Section 3.1 are very important, and they are prototypes for many others of the same type. These examples are characterized by the fact that their "vectors" are actually functions of some type or other. Here are some related examples.
EXAMPLE2: FunG::(S).
We can make the set of allcomplex-valued functions on a nonempty set S, Le., FunG::(S)= {f :S~ C}, into a complex vector space by setting
(f+g)(s)=f(s)+g(s),
(cf)(s) =c(f(sằ
for allf,g EFunG::(S),s ES, and C EC.
EXAMPLE 3: P(lR).
The simplest way to obtain a space akin to but different fromPn(lR)is simply to remove the restriction that the polynomials have degree at mostn. In this way we obtain the vector spaceP(lR)whose vectors are the polynomials
p(x)=ao+aIX+ ... +amxm
with no bound onm. The interpretation of the basic terms we propose in this example is:
vector <-->polynomial,
vector addition <-->addition of polynomials, scalar multiplication <-->multiplication of a polynomial
by a number.
2The multiplication of complex numbers enjoys several important properties, among which are:
(1) the commutative law for multiplication c' c" =c" c' holds;
(2) for every nonzero complex number c there is an inverse c-1such that cc-1=1, and hence
(3) the cancellation law if c' c" =0 and c' -f.0 then c" =0 holds.
The onlyfinite-dimensional(a term we will define and study in Chapter 6) real vector space with these properties is in fact the complex numbers.
Itis again a routine verification that the vector space axioms are sat- isfied.
REMARK: In P(lR) it is possible to multiply two vectors and again receive a vector in P(lR). However, with this product not every element ofP(m.)has an inverse.
EXAMPLE 4: Let S be a nonempty set and T c S. We denote by Fun(S,T)the set of all functions
f:S--'>m.
such that3
f(s) =0 'ifSET.
The set Fun(S,T) c Fun(S) is a vector space in its own right. For if
f,g EFun(S, T)and sET,then
(f+g)(s)=f(s)+ g(s)=0+0=0, and if rEm., then
(rf)(s)=rf(s) =rã0=0,
so thatf +g EFun(S,T),rfEFun(S,T),and therefore Axioms 1,2, and 5-8, which hold in Fun(S), also hold in Fun(S,T). Finally, note that
oEFun(S,T)so Axiom 3 holds, and since(-I)f=-fis also in Fun(S,T) wheneverf EFun(S,T),we see that Axiom 4 holds as well.
EXAMPLE 5: C(a, b).
This is a very fancy example and is included only to indicate the wealth of possible examples of vector spaces.
Let a and b be real numbers with a <b. The vectors ofC(a, b) are the continuous functions defined fora :5;x:5;b. Addition ofvectors is to be addition of functions. That is, iff andg are functions defined and continuous fora :5;x :5; bthenf +g is the function defined by
(f+g)(x)=f(x)+g(x)
for alla:5;x :5;b. Itis an important theorem of the calculus thatf +g is again a continuous function fora:5;x:5;b. Scalar multiplication is to be defined as the ordinary product of a function by a number. If the function is continuous then its product by a number is also continuous.
3The symbol V is an abbreviation for "for all", e.g., VSET means for all elementsS in the setT. If the setT is empty, there are no suchs,and hence no condition.
30 3. Examples of Vector Spaces
The basic terms of a vector space are to be interpreted as follows in this example:
vector +-+continuous function ona $ x $ b, vector addition +-+ addition of functions,
scalar multiplication +-+multiplication of a function by a number.
Again the vector space axioms are easily verified.
Finally we will close this introduction to examples of vector spaces by describing a rather artificial example.
EXAMPLE 6: Let'l!be the set ofallpositivereal numbers and define forA, B E'l!avector sumby
A+B=A.B,
where the product on the right is the usual product of numbers. Ifais a number andA E'l!define
that is the numberA raised to the a-th power. Note that sinceA >0 the so isAa. For example, with these definitions
2+3=6, 2ã 3=9.
We claim that with these definitions ofvector, vector addition and scalar multiplication'l!becomes a vector space. The details ofverification are left to you.
The preceding list only barely scratches the surface of the enormous variety of examples of vector spaces. More examples will appear as we progress through the book and will by no means exhaust the possibili- ties.