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B What do "linear" and "abstract" stand for?

  1. Jun 7, 2017 #1
    What does "linear" in linear algebra and "abstract" in abstract algebra stands for ?

    Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##.

    I don't have any idea about abstract algebra.
     
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  3. Jun 7, 2017 #2

    PeroK

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  4. Jun 7, 2017 #3

    fresh_42

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    It is about linearity in general: linear equations, linear mappings, linear spaces, linear groups: ##M(ax+by)=aM(x)+bM(y)## with scalars, aka numbers ##a,b## and objects like vectors or functions as ##x,y## and ##M## some machinery like a mapping.

    Abstract algebra is a specific English wording as the word "algebra" is in English often used as synonym for "simple calculations". Therefore there is a need for an adjective to distinguish the two. Abstract algebra deals with everything that has multiplications and/or additions: groups, rings, algebras, fields and related objects. Even linear and multilinear stuff like scalar products, tensors and wedge products can be summarized by "abstract algebra", but more often they are - although multilinear - within the context of linear algebra. So "abstract algebra" is more general and with the emphasis on multiplicative structures. E.g. an algebra is a vector space with a multiplication of its vectors.

    As a word that is used informally to describe structures which are not vector spaces in the first place, it is not exactly defined and different authors may use different meanings depending on their purposes as many fields are related.
     
  5. Jun 7, 2017 #4
    So Linear algebra is a small part of Abstract algebra which is more general ?
     
  6. Jun 7, 2017 #5

    fresh_42

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    Strictly speaking, yes. But normally, no. However there are many touching parts and you need methods of linear algebra in ring and field theory, even in group theory. However, usually people distinguish them: vector spaces = linear algebra, abstract algebra = algebraic structures without pure vector spaces. The point is, you cannot always distinguish between the two, but I wouldn't expect much about linear algebra in a book that is titled "Abstract Algebra". It probably presumes you know the linear part already.

    E.g.: I have a book here which is titled "Linear Algebra". It has 444 pages. Imagine this would be part of a book about "Abstract Algebra", too. You would need several volumes. Without further information, I would expect group theory and field theory in a book about abstract algebra. Ring theory, Algebras and other fields I would already search in books titled "Commutative Algebra", "Lie Theory", "Algebraic Geometry", "Associative Algebras", "Applications of Algebra", "Quadratic Forms" or whatever. You see, it is a vast field which could fall under "Algebra".
     
  7. Jun 7, 2017 #6

    mathwonk

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    there is an element of commutativity present in linear algebra that is lacking in more general abstract algebra. Linear algebra concerns the study of commutative rings acting on abelian groups. A principal tool is the Euclidean algorithm for diagonalizing matrices by elementary operations. More general non abelian groups are studied by finding sets they act on, such as subgroups or cosets of subgroups. There is some overlap in that important examples of non abelian groups are provided by groups of invertible matrices, over both finite and infinite fields, such those of determinant one, or those that preserve some notion of length.

    abstract algebra refers in general to the study of algebraic objects by means of the axioms they satisfy, and the properties they have, rather than by direct calculation and manipulation. E.g. the skill of factoring specific polynomials is concrete algebra, but the theorem that every polynomial, (in one or more variables), over a field, can theoretically be factored "uniquely" (i.e. up to unit factors) into irreducible factors is a theorem of abstract algebra. The proof gives no method for actually factoring a specific polynomial.

    Linear algebra also can be studied in an abstract way as well as in a computational concrete way. So linear algebra is not always a branch of abstract algebra. Practicing operations on matrices is computational linear algebra, while proving theorems about dimension and structure of linear operators up to similarity is abstract linear algebra. Almost everything I write about linear algebra, or any other topic, is in the abstract vein, but I do occasionally try to illustrate what is going on with some computational examples. Recently i realized that many of the results I focused on do not allow computations except in artificially simple cases and I rewrote my linear algebra notes to reflect an appreciation for what can be actually computed by hand and what cannot. E.g. the rational canonical form of a matrix can always be computed by hand, say over the rationals, but the Jordan form cannot in general, in any reasonable amount of time at least.

    In most mathematics courses, e.g. calculus as well as linear algebra, the one thing that cannot actually be carried out in practice is to factor polynomials into irreducible factors, but we usually ignore this unpleasant fact and give as examples only cooked ones that do factor easily. Outside the course, and faced with real life examples you are on your own, and may find them quite intractable. There is actually an algorithm for factoring polynomials over the rationals into irreducibles, but the number of steps is absurdly high and hence the process is quite unrealistic.
     
    Last edited: Jun 7, 2017
  8. Jun 7, 2017 #7

    FactChecker

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    Abstract algebra doesn't worry so much about what a math entity represents but it concentrates on what operations are defined for it and what can be said about it because of those operations. Since you are studying linear algebra, you have certainly defined matrices. Matrices can be added. subtracted, and multiplied, but not necessarily divided. Matrix addition is commutative but multiplication is not. Abstract algebra would study what can be said about something that has operators with those properties.

    You might ask what advantage one gets from abstraction. Since math is "the language of science", there are innumerable applications of it with different properties. It's good to ask what can be said about something just because of it's math properties without considering what it represents.
    Some examples:
    Matrices: can be added, subtracted, and multiplied, but not always divided (unless the denominator is invertable). Addition is commutative but multiplication is not.
    Polynomials: can be added, subtracted, and multiplied, but not divided unless the denominator is a factor of the numerator (otherwise division gives a rational function, not a polynomial). Addition and multiplication are commutative.
    Rotations in 2 dimensional space: can be added and subtracted (just consecutive rotations). Addition is commutative.
    Rotations in 3 dimensional space: can be added and subtracted. Addition is not commutative.
    Number systems of different types: integer, real, complex, binary, base n, etc. etc. etc.
    The list of examples goes on and on.
    Abstract algebra classifies the different types of entities and studies what can be said about each type without considering what they represent or are being used for.
     
  9. Jun 8, 2017 #8
    I like the name. I wonder how it got such a name.

    I guess what primes is to number theory is polynomials is to algebra.


    What are the practical use of Abstract algebra ? Like in Physics, Chemistry, Computer science etc ?

    I know that linear algebra is used everywhere. Probably it is as practically useful as Calculus or geometry.
     
  10. Jun 8, 2017 #9

    fresh_42

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    From Sophus Lie. (He / it is pronounced "Lee".)

    Before you ask: Galois theory got its name form Évariste Galois. He has an interesting biography by the way.
     
    Last edited: Jun 8, 2017
  11. Jun 8, 2017 #10
    Yes I know Galois was killed in a duel.
    I think he deserves as much popularity as Newton, Einstein, Ramanujan, Gauss and others. He did so much before the age of 20.
     
  12. Jun 8, 2017 #11

    FactChecker

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    It's probably more accurate to say that primes are to number theory as irreducible polynomials are to a polynomial ring.
    I don't really know a lot about it. I do know that group theory is important for studying the symmetry operations of crystal lattice structures and that breaking codes requires abstract algebra of some type.
    You are right. Linear algebra has so many powerful results that it is used everywhere. When dealing with non-linear things, they are usually locally "linearized" so that linear algebra can be applied locally.
     
  13. Jun 8, 2017 #12

    fresh_42

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    Cryptography, code theory, genetics, theory of formal languages, algorithms, recursive functions, electrical circuits, automatons, sociology, biology, probably computer graphics, AI, and I certainly have forgotten some. As algebra deals with basic structures and simple operations like addition and multiplication, it is as fundamental as linear algebra is. The difficulty with your question is how you define "practical use". You use groups all the time, even if you don't think you do. So it's a natural desire to investigate the underlying patterns.
     
  14. Jun 8, 2017 #13

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    I think that a lot of the value of abstract algebra is that when you are dealing with a mathematical concept, you know what basic questions you should ask (are operations commutative, are there multiplicative inverses, etc.) and what the consequences of the answers are. So you know immediately what you can be confident of and what you can not count on. Almost anything mathematical can be classified as some kind of abstract algebra object.

    PS. I don't think that I have expressed this idea very professionally, but I hope that the idea got through.
     
    Last edited: Jun 9, 2017
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