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- MA4310: Abstract Algebra
- Abstract Algebra Theory and Applications
- Abstract Algebra Theory and Applications
- Linear Algebra, Theory And Applications
MA4310: Abstract Algebra
It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however. I think that the subject of linear algebra is likely the most significant topic discussed in undergraduate mathematics courses. Part of the reason for this is its usefulness in unifying so many different topics. Linear algebra is essential in analysis, applied math, and even in theoretical mathematics.
This is the point of view of this book, more than a presentation of linear algebra for its own sake. This is why there are numerous applications, some fairly unusual. This book features an ugly, elementary, and complete treatment of determinants early in the book. Thus it might be considered as Linear algebra done wrong. I have done this because of the usefulness of determinants. However, all major topics are also presented in an alternative manner which is independent of determinants.
The book has an introduction to various numerical methods used in linear algebra. This is done because of the interesting nature of these methods. The presentation here emphasizes the reasons why they work. It does not discuss many important numerical considerations necessary to use the methods effectively. These considerations are found in numerical analysis texts. This means you ought to have a look at the exercise above it.
Some exercises develop a topic sequentially. There are also a few exercises which appear more than once in the book. I have done this deliberately because I think that these illustrate exceptionally important topics and because some people don't read the whole book from start to finish but instead jump in to the middle somewhere. There is one on a theorem of Sylvester which appears no fewer than 3 times.
Then it is also proved in the text. There are multiple proofs of the Cayley Hamilton theorem, some in the exercises. Some exercises also are included for the sake of emphasizing something which has been done in the preceding chapter. Sometimes a rule specifies a set. For example you could specify a set as all integers larger than 2. It is sometimes said that "A is contained in B" or even "B contains A". The union of two sets is the set consisting of everything which is an element of at least one of the sets, A or B.
Be sure you understand that something which is in both A and B is in the union. It is not an exclusive or. The intersection of two sets, A and B consists of everything which is in both of the sets.
These sorts of sets of real numbers are called intervals. The two points a and b are called endpoints of the interval. In general, the curved parenthesis indicates the end point it sits next to is not included while the square parenthesis indicates this end point is included. Therefore, they cannot be included in any set of real numbers. Mathematicians like to say the empty set is a subset of every set. Set notation is used whenever convenient.
FunctionsThe concept of a function is that of something which gives a unique output for a given input. This is sometimes referred to as the image of f. When this set equals R, the function f is said to be onto, also surjective.
Ordered fieldsThe real numbers R are an example of an ordered field. More generally, here is a definition. Definition 1. With this definition, the familiar properties of order can be proved. The following proposition lists many of these familiar properties. Proposition 1. Proof: First consider 1, called the transitive law. The Complex NumbersJust as a real number should be considered as a point on the line, a complex number is considered a point in the plane which can be identified in the usual way using the Cartesian coordinates of the point.
Thus a, b identifies a point whose x coordinate is a and whose y coordinate is b. You should prove the following theorem. Theorem 1.
The field of complex numbers is denoted as C. An important construction regarding complex numbers is the complex conjugate denoted by a horizontal line above the number. It is defined as follows. What it does is reflect a given complex number across the x axis. Algebraically, the following formula is easy to obtain. With this definition, it is important to note the following. Be sure to verify this.
It is not too hard but you need to do it. For example, consider the distance between 2,5 and 1,8. Complex numbers, are often written in the so called polar form which is described next. A fundamental identity is the formula of De Moivre which follows. Suppose it is true for n. Example 1. The ability to find k th roots can also be used to factor some polynomials. First find the cube roots of The real and complex numbers both are fields satisfying the axioms on Page 13 and it is usually one of these two fields which is used in linear algebra.
The numbers are often called scalars. However, it turns out that all algebraic notions work for any field and there are many others. For this reason, I will often refer to the field of scalars as F although F will usually be either the real or complex numbers. If there is any doubt, assume it is the field of complex numbers which is meant. The reason the complex numbers are so significant in linear algebra is that they are algebraically complete.
Later in the book, proofs of the fundamental theorem of algebra are given. However, here is a simple explanation of why you should believe this theorem. Denote by C r the circle of radius r in the complex plane which is centered at 0. This closed curve will not look like the one in the above diagram or course. The above is just to be a diagram to illustrate what is happening. In fact, the closed curve will typically have all sorts of places where it crosses itself. Now shrink r.
Eventually, for r small enough, the non constant terms are negligible and so A r is a curve which is contained in some circle centered at a 0 which has 0 in its outside. This is one of those arguments which seems all right until you think about it too much. Nevertheless, it will suffice to see that the fundamental theorem of algebra is at least very plausible. A complete proof is in an appendix. Does this formula continue to hold for all integers, n, even negative integers?
Now using De Moivre's theorem, derive a formula for sin 5x and one for cos 5x. Hint: Use the binomial theorem. In words this says the conjugate of a product equals the product of the conjugates and the conjugate of a sum equals the sum of the conjugates. Here is why.
Abstract Algebra Theory and Applications
JudsonStephen F. A current version can always be found text is intended for a one- or two-semester undergraduate course inabstract Algebra. Traditionally, these courses have covered the theoreticalaspects of groups, rings, and fields. Preface This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical. Abstracts , Algebra , Abstract algebra.
Download Annual Edition. From the links below you can download a PDF version of Abstract Algebra: Theory and Applications and the corresponding PreTeXt.
Abstract Algebra Theory and Applications
The HTML version of the annual edition is now available. Information on the print versions to follow soon. If you find an error in the textbook, please email me directly.
Everything you wanted to know about abstract algebra, but were afraid to buy. Note: The Annual Edition has been finalized. See the note about the various Editions and changes.
Exercise solutions to Thomas W. Stevie Bowden. Download PDF. This paper.
Abstract Algebra: Theory and Applications is an open-source textbook written by Tom Judson that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. This text is intended for a one- or two-semester undergraduate course in abstract algebra and covers the traditional theoretical aspects of groups, rings, and fields.
Linear Algebra, Theory And Applications
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. Judson Published Mathematics. Preliminaries The integers Groups Cyclic groups Permutation groups Cosets and Lagrange's theorem Isomorphisms Algebraic coding theory Homomorphisms ad factor groups Matrix groups and symmetry The structure of groups Group actions The sylow theorems Rings Polynomials Integral domains Lattices and boolean algebras Vector spaces Fields Finite fields Introduction to cryptography Galois theory. Save to Library.
Whitehead's Geometric Algebra by Stephen Blake, , pages, 1. Abstract Algebra by John A. Beachy, William D. Blair, Abstract Algebra Wikibooks, Abstract Algebra I by Marcel B. Finan, , PDF.
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