Vector Spaces And Modules



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

1. ;

2. ;

3. ;

4. ;

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.



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

1. ;

2. ;

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

1.   for

2.   for

3.   for

4. If then implies that


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



then



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

1.

2.

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



where

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



to be



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


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.