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Courses Is proof based Linear Algebra be similar to Abstract Algebra

  1. Aug 6, 2017 #1
    I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract Algebra.

    Here are the course descriptions for each class:

    Linear Algebra
    Chapter 1 Vector Spaces
    1.1 Introduction
    1.2 Vector Spaces
    1.3 Subspaces
    1.4 Linear Combinations and Systems of Linear Equations
    1.5 Linear Dependence and Linear Independence
    1.6 Bases and Dimension

    Chapter 2 Transformations and Matrices
    2.1 Linear Transformations, Null Spaces, and Ranges
    2.2 The Matrix Representation of a Linear Transformation
    2.3 Composition of Linear Transformations and Matrix Multiplication
    2.4 Invertibility and Isomorphisms
    2.5 The Change of Coordinates Matrix

    Chapter 3 Elementary Matrix Operations and Systems of Linear Equations
    3.1 Elementary Matrix Operations and Elementary Matrices
    3.2 The Rank of a Matrix and Matrix Inverses
    3.3 Systems of Linear Equations – Theoretical Aspects
    3.4 Systems of Linear Equations – Computational Aspects
    Index of Definitions

    Chapter 4 Determinants
    4.4 Summary – Important Facts about Determinants

    Chapter 5 Diagonalization
    5.1 Eigenvalues and Eigenvectors
    5.2 Diagonalizability

    Chapter 6 Inner Product Spaces
    6.1 Inner Products and Norms
    6.2 The Gram-Schmidt Orthogonalization Process

    Chapter 7 Canonical Forms
    7.1 The Jordan Canonical Form I
    7.2 The Jordan Canonical Form II
    7.3 The Minimal Polynomial

    Abstract Algebra I
    A. Group Theory
    Binary Operations
    Groups
    Subgroups
    Permutation Groups
    Orbits and Cycles
    Cyclic Groups
    Cosets and Lagrange
    Homomorphisms
    Isomorphisms and Cayley’s Theorem
    Factor Groups
    Fundamental Homomorphism Theorem

    B. Rings
    Rings and Fields
    Integral Domains
    Little Fermat and Euler Theorems
    Fields of Quotients
    Polynomial Rings
    Polynomial and Division Algorithm
    Remainder Theorem/Factor Theorem
    Homomorphisms and Factor Rings
    Prime and Maximal Ideals and PIDs, Prime Ideals

    C. Field Theory (Introduction)
    Field Extensions
    Existence of a Splitting Field
    Constructibility with Ruler and Compass –Trisecting Angles

    I'm currently taking a course called "Matrix Algebra", which covers things from Linear Algebra such as Matrices, determinants, vector spaces, eigenvalues, orthogonality, but it's mainly computations based while the Linear Algebra course above is more proof based, proving theorems then computations. But are the proofs similar to Abstract Algebra and how hard would a proof based Linear Algebra course be compared to the first part of Abstract Algebra? Only curious because many people say Abstract Algebra is more difficult than any Algebra course because it's very proof oriented while a Linear Algebra course is more applied (there are proofs in my Matrix Algebra course, but my professor is not going to make us prove anything on a test, just applications and calculations).
     
  2. jcsd
  3. Aug 6, 2017 #2

    fresh_42

    Staff: Mentor

    Well, you can present both either way. I would say the difficulty level is similar. For a physicist linear algebra might be more important, as it mainly deals with coordinates and coordinate transformations, which are especially important in physics. Abstract algebra on the other hand deals more with the related structures. They might also become important in a physicist's life, but usually later on in cosmology or quantum field theory. In algebra there are also a lot of examples and calculations.

    And last but not least, it always depends on personal tastes which one is felt more difficult. The proofs themselves are actually a bit different, but not necessarily of different difficulty, the more as there are many sections which require knowledge from both.
     
  4. Aug 6, 2017 #3

    Thanks for your reply. So essentially a proof based Linear Algebra course would be the same level of difficulty as an Abstract Algebra and the proofs, though different, are not so difficult.

    As for physics, I would think proof based math classes like Abstract Algebra are not that useful in Physics unless it were theoretical physics or something (i've heard analysis may be useful, not sure on that)? From what I researched, most physics majors take Calculus I-II-III, Differential Equations and Linear Algebra and don't take any math courses beyond that (unless they want to). Kind of makes sense since Calculus D.E. and LA are more applied math so it works for Physics. And there are some colleges I've looked into that only require Calculus as a math prerequisite for Physics and nothing else.
     
  5. Aug 6, 2017 #4

    fresh_42

    Staff: Mentor

    I'd say so. Depends a bit on a person's ability of abstraction. Linear algebra is more ground based (as often things can be drawn or sketched), and abstract algebra is a bit more axiomatic.
    Not sure whether experimental and theoretical physics should be divided at such an early state. Almost certainly not. The importance of proofs may be downstream, except that they teach how to use the methods. The concepts and their objects, such as groups, rings or modules are not. It is where physics takes place. Not everything in physics is real, complex and a finite dimensional vector space. But many of it will appear naturally on your way, without you noticing which area they're from. However, there are other important concepts like
    analysis, which is extremely important. I'm still not sure where to draw the line between analysis and calculus, so I use them more or less synonymous in English.
    I agree, that this is the minimal equipment and as always, everything depends on how far and even more important, how deep you want to go. I would add some parts of functional analysis (deals with operators) and topology (for modern cosmology) to the list, and maybe more. But this depends on how far ... oops, already mentioned.
    Calculus: real, vector, complex is essential for physics. No way to get around. Modern physics started basically with Newton, and him and Leibniz both felt the need to introduce differentiation - et voilà: calculus is born. O.k. this was eventually a bit too short of a description and many important concepts have been added since then (and partly earlier), but in brief: that was it.

    Mathematics and physics are siblings with all what it means.
     
  6. Aug 6, 2017 #5
    I haven't really went deep into physics because I just started Physics, but definitely plan on taking more advanced physics courses, but from what I researched, people divide physics into experimental and theoretical, the latter being more math. Also, from what I heard, physicists are not concern about how the math itself works, but how it can be used to describe an experiment and/or natural phenomenon.


    Isn't Analysis just rigorous or proof based Calculus, proving the fundamental theorem of Calculus and how Calculus works. I can see how this is very important for Physics. Also, i've heard you can taken Analysis without having any sort of background in Calculus (obviously I have taken Calculus I to III and Differential Equations). Some Colleges have Calculus III as a prerequisite to real/complex analysis, while others only require an introduction to proofs and logic class as a prerequisite for real/complex analysis. Could you take analysis without having Calculus as a prerequisite but obviously knowing how to do proofs?




    Yes, kind of like how for Computer Science, discrete math is the most important math for CS along with many other courses. For Physics, obviously math is the language of Physics, but Calculus is the most important math for Physics. Again, I only started taking Physics (Calculus based intro physics classes which is mostly using Algebra and Trig with basic derivative and integral computation). Linear Algebra I can see though is important for both because of vector properties and all.
     
  7. Aug 7, 2017 #6

    fresh_42

    Staff: Mentor

    In principle: Yes, calculus is calculation, analysis investigation. But I've seen it so often used as synonyms that I gave up to distinguish them (in English). In my language, calculus is used quite differently as being a certain system of objects and rules to do calculations. Analysis (in its mathematical context) is reserved to "calculus" on college level and of course with proofs. An example of calculus would then be: You can prove the fundamental theorem of algebra by several methods: within an analytical calculus, a topological calculus, a functional analysis calculus, a geometric calculus etc. So calculus would refer to the framework and language used. At its heart this is also the meaning in English, but rarely used within these constraints. E.g. you said you have taken calculus I-III, but you only refer to an analytic calculus with continuous or differentiable functions and stuff. And I would be surprised, if e.g. the intermediate value theorem wasn't proven. So it contains proofs and is restricted to a certain analytic framework. Now, what's the difference to analysis? That analysis extends above calculus? That measure theory for example is also within analysis but not within calculus? That would be semantics.
     
  8. Aug 7, 2017 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    there is one sense in which linear algebra and abstract algebra [which for you seems to mean group theory], have a different flavor. Namely the arguments in linear algebra are mainly done by reasoning on polynomials and linear maps, hence have a heavy use of commutativity. Abstract group theory arguments must work with non commutative operations so are a bit different. I.e. in a nutshell, the key to most theoretical results in linear algebra is the fact that a pair consisting of a k-vector space and a single operator, defines a module structure on the vector space over the (commutative) polynomial ring with coefficients in k, hence the Euclidean algorithm can be applied to study the operator.

    In group theory, one cannot consider the group as a module over any commutative ring unless the group is abelian, in which case the theory is exactly parallel to that of linear algebra, the module structure on an abelian group being defined over the integers. The basic tool for studying non commutative groups is the theory of "group actions", especially the action of conjugation by elements of the group acting on subgroups of the group. In the abelian case of course the action by conjugation is trivial.

    Of course matrix multiplication in linear algebra is non commutative, so if you are studying an entire group of linear operators, such as O(n) or GL(n), then group theoretical arguments are also used and there is a non commutative aspect to it. But most courses in theoretical linear algebra discuss only one operator at a time, or at most linear combinations of powers of a single operator, i.e. polynomials in one operator, and these all commute with one another.

    To summarize and repeat, abstract group theory usually studies an arbitrary, often finite, group G. In linear algebra one usually studies a single operator T:V-->V, and the commutative subalgebra k[T] associated to it, inside the (non commutative) algebra End(V).

    The point of this discussion is that to some people, of course not all, commutative algebra may seem easier than non commutative algebra.

    Well I should modify or maybe retract much of the remarks above, after reading more closely the syllabus for your courses. I.e. the only group theory covered there is very elementary, and no group actions seem needed, since they don't even discuss the Sylow theorems. And the second 2/3 of the abstract course is commutative algebra, basic ring and field theory, and only the elementary part of field theory, i.e. no Galois theory hence no non commutative group theory. Moreover the linear algebra course goes all the way to the often difficult discussion of the Jordan canonical form, which in fact may seem harder than any of the basic stuff in the syllabus for abstract algebra. So the linear algebra course includes some "hard" linear algebra while the abstract course includes only "easy" abstract algebra. Just my opinion of course.
     
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