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Finite-Dimensional Vector Spaces

Dover Publications, Inc.

SPACES

§ 1. Fields

In what follows we shall have occasion to use various classes of numbers (such as the class of all real numbers or the class of all complex numbers). Because we should not, at this early stage, commit ourselves to any specific class, we shall adopt the dodge of referring to numbers as scalars. The reader will not lose anything essential if he consistently interprets scalars as real numbers or as complex numbers; in the examples that we shall study both classes will occur. To be specific (and also in order to operate at the proper level of generality) we proceed to list all the general facts about scalars that we shall need to assume.

(A) To every pair, α and β of scalars there corresponds a scalar α + β, called the sum of α and β, in such a way that

(1) addition is commutative, α + β = β + α,

(2) addition is associative, α + (β + γ) = (α + β) + γ,

(3) there exists a unique scalar 0 (called zero) such that α + 0 = α for every scalar α, and

(4) to every scalar α there corresponds a unique scalar -α such that α + (-α) = 0.

(B) To every pair, α and β, of scalars there corresponds a scalar αβ, called the product of α and β, in such a way that

(1) multiplication is commutative, αβ = βα,

(2) multiplication is associative, α(βγ) = (αβ)γ,

(3) there exists a unique non-zero scalar 1 (called one) such that α1 = α for every scalar α, and

(4) to every non-zero scalar α there corresponds a unique scalar α-1 (or 1/α) such that α α-1 = 1.

(C) Multiplication is distributive with respect to addition, α(β + γ) = αβ + αγ.

If addition and multiplication are defined within some set of objects (scalars) so that the conditions (A), (B), and (C) are satisfied, then that set (together with the given operations) is called a field. Thus, for example, the set Q of all rational numbers (with the ordinary definitions of sum and product) is a field, and the same is true of the set (R of all real numbers and the set C of all complex numbers.

EXERCISES

1. Almost all the laws of elementary arithmetic are consequences of the axioms defining a field. Prove, in particular, that if F is a field, and if α, β, and γ belong to F, then the following relations hold.

(a) 0 + α = α

(b) If α + β = α + γ, then β = γ.

(c) α + (β - α = β. (Here β - α = β + (-α).)

(d) α·0 = α·0 = 0. (For clarity or emphasis we sometimes use the dot to indicate multiplication.)

(e) (-1)α = -α.

(f) (-α)(-β) = αβ.

(g) If αβ = 0, then either α = 0 or β = 0 (or both).

2. (a) Is the set of all positive integers a field? (In familiar systems, such as the integers, we shall almost always use the ordinary operations of addition and multiplication. On the rare occasions when we depart from this convention, we shall give ample warning. As for "positive," by that word we mean, here and elsewhere in this book, "greater than or equal to zero." If 0 is to be excluded, we shall say "strictly positive.")

(b) What about the set of all integers?

(c) Can the answers to these questions be changed by re-defining addition or multiplication (or both)?

3. Let m be an integer, m ≤ 2, and let Zm be the set of all positive integers less than m, Zm = {0, 1, ···, m - 1}. If α and β are in Zm, let α + β be the least positive remainder obtained by dividing the (ordinary) sum of α and β by m, and, similarly, let αβ be the least positive remainder obtained by dividing the (ordinary) product of α and β by m. (Example: if m = 12, then 3 + 11 = 2 and 3-11 = 9.)

(a) Prove that Zm is a field if and only if m is a prime.

(b) What is -1 in Z5?

(c) What is 1/3 in Z7?

4. The example of Zp (where p is a prime) shows that not quite all the laws of elementary arithmetic hold in fields; in Z2, for instance, 1 + 1 = 0. Prove that if F is a field, then either the result of repeatedly adding 1 to itself is always different from 0, or else the first time that it is equal to 0 occurs when the number of summands is a prime. (The characteristic of the field F is defined to be 0 in the first case and the crucial prime in the second.)

5. Let Q([square root of 2]) be the set of all real numbers of the form α + β [square root of 2], where α and β are rational.

(a) Is Q([square root of 2]) a field?

(b) What if α and β are required to be integers?

6. (a) Does the set of all polynomials with integer coefficients form a field?

(b) What if the coefficients are allowed to be real numbers?

7. Let F be the set of all (ordered) pairs (α, β) of real numbers.

(a) If addition and multiplication are defined by

(α, β) + (γ, δ) = (α] + γ, β + δ)

and

(α, β) + (γ, δ) = (α]γ, βδ),

does F become a field?

(b) If addition and multiplication are defined by

(α, β) + (γ, δ) = (α] + γ, β + δ)

and

(α, β) + (γ, δ) = (α]γ, βδ, αδ + βγ),

is F a field then?

(c) What happens (in both the preceding cases) if we consider ordered pairs of complex numbers instead?

§ 2. Vector spaces

We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field F; the scalars to be used are to be elements of F.

Definition.A vector space is a set V of elements called vectors satisfying the following axioms.

(A) To every pair, x and y, of vectors in V there corresponds a vector x + y, called the sum of x and y, in such a way that

(1) addition is commutative, x + y = y + x,

(2) addition is associative, x + (y + z) = (x + y) + z,

(3) there exists in V a unique vector 0 (called the origin) such that x + 0 = x for every vector x, and

(4) to every vector x in V there corresponds a unique vector -x such that x + (-x) = 0.

(B) To every pair, α and x, where α is a scalar and a; is a vector in V, there corresponds a vector ax in V, called the product of α and x, in such a way that

(1) multiplication by scalars is associative, α(βx) = (αβ)x, and

(2) 1x = x for every vector x.

(C) (1) Multiplication by scalars is distributive with respect to vector addition, a(x + y) = ax + ay, and

(2) multiplication by vectors is distributive with respect to scalar addition, (α + β)x = αx + βx.

These axioms are not claimed to be logically independent; they are merely a convenient characterization of the objects we wish to study. The relation between a vector space V and the underlying field F is usually described by saying that V is a vector space over F. If F is the field (R of real numbers, V is called a real vector space; similarly if F is Q or if F is C, we speak of rational vector spaces or complex vector spaces.

§ 3. Examples

Before discussing the implications of the axioms, we give some examples. We shall refer to these examples over and over again, and we shall use the notation established here throughout the rest of our work.

(1) Let C1 (= C) be the set of all complex numbers; if we interpret x + y and αx as ordinary complex numerical addition and multiplication, C1 becomes a complex vector space.

(2) Let (P be the set of all polynomials, with complex coefficients, in a variable t. To make (P into a complex vector space, we interpret vector addition and scalar multiplication as the ordinary addition of two polynomials and the multiplication of a polynomial by a complex number; the origin in (P is the polynomial identically zero.

Example (1) is too simple and example (2) is too complicated to be typical of the main contents of this book. We give now another example of complex vector spaces which (as we shall see later) is general enough for all our purposes.

(3) Let Cn, n = 1, 2, ···, be the set of all n-tuples of complex numbers. If x = ([xi]1, ···, [xi]1) and y = (η1 ··· ηn are elements of Cn, we write, by definition,

[MATHEMATICAL EXPRESSION OMITTED]

It is easy to verify that all parts of our axioms (A), (B), and (C), § 2, are satisfied, so that Cn is a complex vector space; it will be called n-dimensional complex coordinate space.

(4) For each positive integer n, let Pn be the set of all polynomials (with complex coefficients, as in example (2)) of degree ≤n -1, together with the polynomial identically zero. (In the usual discussion of degree, the degree of this polynomial is not defined, so that we cannot say that it has degree ≤n -1.) With the same interpretation of the linear operations (addition and scalar multiplication) as in (2), Pn is a complex vector space.

(5) A close relative of Cn is the set Rn of all n-tuples of real numbers. With the same formal definitions of addition and scalar multiplication as for Cn, except that now we consider only real scalars a, the space Rn is a real vector space; it will be called n-dimensional real coordinate space.

(6) All the preceding examples can be generalized. Thus, for instance, an obvious generalization of (1) can be described by saying that every field may be regarded as a vector space over itself. A common generalization of (3) and (5) starts with an arbitrary field F and forms the set Fn of n-tuples of elements of F; the formal definitions of the linear operations are the same as for the case F = C.

(7) A field, by definition, has at least two elements; a vector space, however, may have only one. Since every vector space contains an origin, there is essentially (i.e., except for notation) only one vector space having only one vector. This most trivial vector space will be denoted by 0.

(8) If, in the set R of all real numbers, addition is defined as usual and multiplication of a real number by a rational number is defined as usual, then R becomes a rational vector space.

(9) If, in the set C of all complex numbers, addition is defined as usual and multiplication of a complex number by a real number is defined as usual, then C becomes a real vector space. (Compare this example with (1); they are quite different.)

§4. Comments

A few comments are in order on our axioms and notation. There are striking similarities (and equally striking differences) between the axioms for a field and the axioms for a vector space over a field. In both cases, the axioms (A) describe the additive structure of the system, the axioms (B) describe its multiplicative structure, and the axioms (C) describe the connection between the two structures. Those familiar with algebraic terminology will have recognized the axioms (A) (in both §1 and §2) as the defining conditions of an abelian (commutative) group; the axioms (B) and (C) (in §2) express the fact that the group admits scalars as operators. We mention in passing that if the scalars are elements of a ring (instead of a field), the generalized concept corresponding to a vector space is called a module.

Special real vector spaces (such as R2 and R3 are familiar in geometry. There seems at this stage to be no excuse for our apparently uninteresting insistence on fields other than R, and, in particular, on the field C of complex numbers. We hope that the reader is willing to take it on faith that we shall have to make use of deep properties of complex numbers later (conjugation, algebraic closure), and that in both the applications of vector spaces to modern (quantum mechanical) physics and the mathematical generalization of our results to Hilbert space, complex numbers play an important role. Their one great disadvantage is the difficulty of drawing pictures; the ordinary picture (Argand diagram) of C1 is indistinguishable from that of R2, and a graphic representation of C2 seems to be out of human reach. On the occasions when we have to use pictorial language we shall therefore use the terminology of Rn in Cn, and speak of C2, for example, as a plane.

Finally we comment on notation. We observe that the symbol 0 has been used in two meanings: once as a scalar and once as a vector. To make the situation worse, we shall later, when we introduce linear functionals and linear transformations, give it still other meanings. Fortunately the relations among the various interpretations of 0 are such that, after this word of warning, no confusion should arise from this practice.

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Excerpted from Finite-Dimensional Vector Spaces by Paul R. Halmos. Copyright © 2017 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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