Vector Spaces And Modules
Up to this point we have been introduced to groups and to rings; the former has its motivation in the set of one-to-one mappings of a set onto itself, the latter, in the set of integers. The third algebraic model which we are about to consider --- vector space --- can, in large part, trace its origins to topics in geometry and physics.
Its description will be reminiscent of those of groups and rings --- in fact, part of its structure is that of an abelian group --- but a vector space differs from these previous two structures in that one of the products defined on it uses elements outside of the set itself. These remarks will become clear when we make the definition of a vector space.
Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. For this reason the basic concepts introduced in them have a certain universality and are ones we encounter, and keep encountering, in so many diverse contexts. Among these fundamental notions are those of linear dependence, basis, and dimension which will be developed in this chapter. These are potent and effective tools in all branches of mathematics; we shall make immediate and free use of these in many key places in Chapter 5 which treats the theory of fields.
Intimately intertwined with vector spaces are the homomorphisms of one vector space into another (or into itself). These will make up the bulk of the subject matter to be considered in Chapter 6.
In the last part of the present chapter we generalize from vector spaces to modules; roughly speaking, a module is a vector space over a ring instead of over a field. For finitely generated modules over Euclidean rings we shall prove the fundamental basis theorem. This result allows us to give a complete description and construction of all abelian groups which are generated by a finite number of elements.
Elementary Basic Concepts
DEFINITION: A nonempty set is said to be a vector space over a field if is an abelian group under an operation which we denote by and if for every there is defined an element, written in subject to
for all (where the 1 represents the unit element of under multiplication).
Note that in Axiom 1 above the is that of whereas on the left-hand side of Axiom 2 it is that of and on the right-hand side, that of
We shall consistently use the following notations:
(a) will be a field.
(b) Lowercase Greek letters will be elements of ; we shall often refer to elements of as scalars.
(c) Capital Latin letters will denote vector spaces over
(d) Lowercase Latin letters will denote elements of vector spaces. We shall often call elements of a vector space vectors.
If we ignore the fact that has two operations defined on it and view it for a moment merely as an abelian group under Axiom 1 states nothing more than the fact that multiplication of the elements of by a fixed scalar defines a homomorphism of the abelian group into itself. From Lemma 4.1.1 which is to follow, if this homomorphism can be shown to be an isomorphism of onto
This suggests that many aspects of the theory of vector spaces (and of rings, too) could have been developed as a part of the theory of groups, had we generalized the notion of a group to that of a group with operators. For students already familiar with a little abstract algebra, this is the preferred point of view; since we assumed no familiarity on the reader's part with any abstract algebra, we felt that such an approach might lead to a too sudden introduction to the ideas of the subject with no experience to act as a guide.
Example 4.1.1: Let be a field and let be a field which contains as a subfield. We consider as a vector space over using as the of the vector space the addition of elements of and by defining, for to be the products of and as elements in the field Axioms 1, 2, 3 for a vector space are then consequences of the right-distributive law, left-distributive law, and associative law, respectively, which hold for as a ring.
Example 4.1.2: Let be a field and let be the totality of all ordered -tuples, where the Two elements and of are declared to be equal if and only if for each We now introduce the requisite operations in to make of it a vector space by defining:
It is easy to verify that with these operations, is a vector space over Since it will keep reappearing, we assign a symbol to it, namely
Example 4.1.3: Let be any field and let the set of polynomials in over We choose to ignore, at present, the fact that in we can multiply any two elements, and merely concentrate on the fact that two polynomials can be added and that a polynomial can always be multiplied by an element of With these natural operations is a vector space over
Example 4.1.4: In let be the set of all polynomials of degree less than Using the natural operations for polynomials of addition and multiplication, is a vector space over F.
What is the relation of Example 4.1.4 to Example 4.1.2? Any element of is of the form where ; if we map this element onto the element in we could reasonably expect, once homomorphism and isomorphism have been defined, to find that and are isomorphic as vector spaces.
DEFINITION: If is a space over and if then is a subspace of if under the operations of itself, forms a vector space over Equivalently, is a subspace of whenever implies that
Note that the vector space defined in Example 4.1.4 is a subspace of that defined in Example 4.1.3. Additional examples of vector spaces and subspaces can be found in the problems at the end of this section.
DEFINITION: If and are vector spaces over then the mapping of into is said to be a homomorphism if
for all and all
As in our previous models, a homomorphism is a mapping preserving all the algebraic structure of our system.
If in addition, is one-to-one, we call it an isomorphism. The kernel of is defined as where is the identity element of the addition in It is an exercise that the kernel of is a subspace of and that is an isomorphism if and only if its kernel is Two vector spaces are said to be isomorphic if there is an isomorphism of one onto the other.
The set of all homomorphisms of into will be written as Hom Of particular interest to us will be two special cases, Hom and Hom We shall study the first of these soon; the second, which can be shown to be a ring, is called the ring of linear transformations on A great deal of our time, later in this book, will be occupied with a detailed study of Hom
We begin the material proper with an operational lemma which, as in the case of rings, will allow us to carry out certain natural and simple computations in vector spaces. In the statement of the lemma, represents the zero of the addition in that of the addition in and the additive inverse of the element of
LEMMA 4.1.1: If is a vector space over then
4. If then implies that
Proof: The proof is very easy and follows the lines of the analogous results proved for rings; for this reason we give it briefly and with few explanations.
1. Since we get
2. Since we get
4. Assume to the contrary that and Then
which is a contradiction.
The lemma just proved shows that multiplication by the zero of or of always leads us to the zero of Thus there will be no danger of confusion in using the same symbol for both of these, and we henceforth will merely use the symbol to represent both of them.
Let be a vector space over and let be a subspace of Considering these merely as abelian groups construct the quotient group ; its elements are the cosets where The commutativity of the addition, from what we have developed in Chapter 2 on group theory, assures us that is an abelian group. We intend to make of it a vector space. If define
As is usual, we must first show that this product is well defined; that is, if
Now, because is in ; since is a subspace, must also be in Using part 3 of Lemma 4.1.1 (see Problem 1) this says that and so Thus
the product has been shown to be well defined. The verification of the vector-space axioms for is routine and we leave it as an exercise. We have shown
LEMMA 4.1.2: If is a vector space over and if is a subspace of then is a vector space over , where, for and
is called the quotient space of by
Without further ado we now state the first homomorphism theorem for vector spaces; we give no proofs but refer the reader back to the proof of Theorem 2.7.1.
THEOREM 4.1.1: If is a homomorphism of onto with kernel then is isomorphic to Conversely, if is a vector space and a subspace of then there is a homomorphism of onto
The other homomorphism theorems will be found as exercises at the end of this section.
DEFINITION: Let be a vector space over and let be subspaces of is said to be the internal direct sum of if every element can be written in one and only one way as
Given any finite number of vector spaces over consider the set of all ordered -tup1es where We declare two elements and of to be equal if and only if for each . We add two such elements by defining
Finally, if and we define to be
To check that the axioms for a vector space hold for with its operations as defined above is straightforward. Thus itself is a vector space over We call the external direct sum of and denote it by writing
THEOREM 4.1.2: If V is the internal direct sum of then is isomorphic to the external direct sum of
Proof: Given can be written, by assumption, in one and only one way as
where ; define the mapping of into by
Since has a unique representation of this form, is well defined. It clearly is onto, for the arbitrary element
We leave the proof of the fact that is one-to-one and a homomorphism to the reader.
Because of the isomorphism proved in Theorem 4.1.2 we shall henceforth merely refer to a direct sum, not qualifying that it be internal or external.