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By Charles G. We will begin by discussing two familiar problems which will serve as motivation for much of what will follow. First of all, you are all familiar with the problem of finding the solution or solutions of a system of linear equations. For example the system.
Most of the techniques you have learned for finding solutions of systems of this type become very unwieldy if the number of unknowns is large or if the coefficients are not integers. It is not uncommon today for scientists to encounter systems like 1. Even using the most efficient techniques known, a fantastic amount of arithmetic must be done to solve such a system. The development of high-speed computational machines in the last 20 years has made the solution of such problems feasible.
Using the elementary analytical geometry of three-space, one can provide a convenient and fruitful geometric interpretation for the system 1.
Since each of the three equations represents a plane, we would normally expect the three planes to intersect in precisely one point, in this case the point with coordinates 1, — 1, 1. Our geometric point of view suggests that this will not always be the case for such systems since the following two special cases might arise:.
Two of the planes might be parallel, in which case there would be no points common to the three planes and hence no solution of the system. The planes might intersect in a line and hence there would be an infinite number of solutions. The first of these special cases could be illustrated in the system obtained from 1. Note, by the way, the advantages of the double subscript notation in writing the general linear system 1. It may be difficult for you to interpret this system geometrically as we did with 1.
The second familiar problem we shall mention is that of finding the principal axes of a conic section. The curve defined by. In other words, there is a coordinate system which in some sense is most natural or most convenient for investigating the curve defined by 1.
This problem becomes very complicated—even in three dimensions—if one uses only elementary techniques. Later on we will discover efficient ways of finding the most convenient coordinate system for investigating the quadratic function. We will find it convenient to use a small amount of standard mathematical shorthand as we proceed. Learning this notation is part of the education of every mathematician.
In addition the gain in efficiency will be well worth any initial inconvenience. A set of objects can be specified by listing its members or by giving some criteria for membership. For example,. We will commonly use this set-builder notation in describing sets. Thus we would read. If A and B are any two sets, then the intersection of A and B is defined to be. In dealing with sets and relationships between sets, it is often convenient to construct simple pictures such as those in Fig.
In the following pages, we will frequently need to show that two sets, A and B , are equal. This is almost always handled by proving that A is a subset of B and that B is also a subset of A. The procedure can be represented schematically by. It is clear that an inductive set T contains every positive integer.
One can frequently prove theorems about all the positive integers by showing that the set of integers for which the theorem is true the truth set of the theorem is an inductive set. This proof technique is called mathematical induction and will be used on occasion in this book. Draw a Venn diagram to represent.
Addition and multiplication are essentially binary compositions , that is, they are compositions which produce a new number the sum or product from two other numbers. Note that not all binary compositions are associative. Another important example of a nonassociative composition with which many readers will be familiar is the vector cross product. Life is full of noncommutative compositions: for example, every chemistry student knows that the result of addition mixing of chemical compounds depends on order.
If one adds concentrated sulfuric acid to water one gets dilute sulfuric acid, but adding water to sulfuric acid is likely to produce an explosion. As to less academic things, you all know that the order in which things are done on a date makes a great deal of difference in the end results.
Addition and multiplication, the two main binary compositions for the real numbers, are connected by. The numbers 0 and 1 occupy special places in the real number system. They are known as identity elements for addition and multiplication, respectively. The essential algebraic properties of these elements are given below. The last of the essential algebraic properties of the real numbers concern inverse elements and provide us with what is needed to define operations inverse to addition and multiplication, that is, subtraction and division.
There are many more algebraic properties of the real numbers but they are all consequences of the nine properties listed above. These nine properties do not, however, completely determine the real number system since, for example, the set of complex numbers also satisfies these properties. We now take this generalization process one giant step further with.
Definition 1. These nine properties postulates are commonly used to define other types of abstract systems as well. The study of such systems is in part the subject matter of a course in modern algebra and will not be pursued here at any great length.
Figure 1. We have introduced the notion of a field here, since most of the results we obtain will be valid over any field. Let us first look at some examples of fields. Example 1. Example 2. Example 3. If the reader thinks that this is a useless example, he should discuss the matter with a person who understands the design of digital computers. Example 5 x ,. All of these examples are familiar, except possibly Example 4, and we will leave it to the reader to verify, in detail, that each of these systems satisfies the nine properties of Definition 1.
Theorem 1. Proof is a subfield, then the only if assertion is clear. This establishes Properties 2, 3, 7, and 8 of is closed under multiplication. Is this composition associative? Directly from Definition 1. The remainder of this chapter is concerned principally with matrices and their relationship to systems of linear algebraic equations. The first order of business is to formally define the term matrix. Two matrices are equal if and only if they are identical.
Note again that the row index is always listed first. The double subscript notation for the elements permits us to discuss the elements or entries of a general matrix in an efficient manner. Note that the first subscript on the entry mij indicates the row in which the entry occurs while the second subscript indicates the column in which it occurs.
The ordered pair i , j is called the address of the entry mij , and the entry in the i , j position of M is mij. The entry mij will be said to have row index i and column index j. If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. The use of commas when writing a row matrix like Z is a concession to standard punctuation. Matrices which have only one row row matrices and matrices which have only one column column matrices are of great importance.
In particular, the rows and columns of a general matrix deserve special attention. We will denote the i th row of M by. For emphasis and easy reference, the notational conventions to be used in dealing with matrices are listed below:.
Matrices will normally be designated by capital letters. If a matrix has a name more complicated than a single letter, for example, Row2 M , that name will always begin with a capital letter. If it is necessary to specify the size of a matrix, we will do so directly below the name of the matrix; thus a general r by c matrix M. If a capital letter, say M, represents a matrix, then the corresponding lowercase letter, with double subscripts, will represent an entry of M ; for example, m 23 represents the element of M which is in row 2 and column 3.
Note that we use ent rather than Ent since ent ij M is a scalar, not a matrix. Matrices arise naturally in the study of linear systems like 1. Let us consider the special case of Eq. Note that to make our motivating substitution it is essential that the number of rows of B equals the number of columns of A. There are several things about matrix multiplication which should be emphasized.
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By Charles G. We will begin by discussing two familiar problems which will serve as motivation for much of what will follow. First of all, you are all familiar with the problem of finding the solution or solutions of a system of linear equations. For example the system. Most of the techniques you have learned for finding solutions of systems of this type become very unwieldy if the number of unknowns is large or if the coefficients are not integers. It is not uncommon today for scientists to encounter systems like 1. Even using the most efficient techniques known, a fantastic amount of arithmetic must be done to solve such a system.
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Other titles in the Dover Books on. Introduction to Matrices and Linear Transformations 2nd Finkbeiner. Matrices and Linear Transformations by Charles G. Our US book-manufacturing partners produce the highest. Undergraduate-level introduction to linear algebra and matrix theory deals with matrices and linear systems, vector spaces, determinants, linear transformations.
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