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Algebra I vs Linear Algebra I

  1. May 4, 2005 #1
    Perhaps this belongs better in the career guidance forum (in which case, please move it moderator) but I thought it would be more specific to this forum.

    I'm planning to study physics at the university level sometime in the future. My plan is to work on the first year math credits first, to build a solid foundation. Perhaps part-time.

    With that said, I'm forced to confront "Algebra I" vs "Linear Algebra I". By the names I would assume (yeah, bad idea) that Linear Algebra I is just a subset of Algebra I (but much more specific and indepth for that area).

    There are a few streams of calculus at UofToronto. There is the lower Calculus I/II, then the practical Calculus!/Multivariable Calculus and finally the theoretical approach with Analysis I/Analysis II.

    Analysis II requires Algebra I (the alternative streams require none). The rest of the second, third and fourth year math courses will accept either Algebra I/II or Linear Algebra I/II in the prerequisites if it is there.

    So, as someone who is far removed from these, perhaps someone may enlighten me as to which I should take. Or should I aim to complete both?

    Course descriptions from UofT:

    Linear Algebra I
    Matrix arithmetic and linear systems. Rn: subspaces, linear independence, bases, dimension; column spaces, null spaces, rank and dimension formula. Orthogonality orthonormal sets, Gram-Schmidt orthogonalization process; least square approximation. Linear transformations Rn->Rm. The determinant, classical adjoint, Cramer's Rule. Eigenvalues, eigenvectors, eigenspaces,
    diagonalization. Function spaces and application to a system of linear differential equations.

    Algebra I
    A theoretical approach to: vector spaces over arbitrary fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.

    I'm hoping someone can tell me if the Linear Algebra is inclusive in the Algebra stream, too.

    Like I said above--I've no clue what most of this is and therefore don't know how pertinent it is to my future career (both academic and real world). Any help by you Master Jedis is appreciated by this young padawan :D

    Last edited: May 4, 2005
  2. jcsd
  3. May 4, 2005 #2


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    The designation "Algebra I" in the course you list above is non-standard. Normally, the term "Algebra I" would be applied to the Algebra typically studied at about age 15, "Linear Algebra" would be studied typically by a first or second year college stuent and focus on vectors and matrixes, and "Abstract Algebra" would typically be studied by upper division math and physics majors involving things like group theory.

    In this setting it looks like Algebra and Linear Algebra are the same course, but that Algebra is taught from a more theoretical perspective.

    When I was in college (a decade ago), the physicists took what we called "Applied Analysis" which focused on Fourier Transforms and Tensor mathematics, while the math majors mostly took "Real Analysis" which was basically a very rigorous treatment of calculus from a theoretical perspective.

    These days, it depends on what kind of physicist you want to be. String theory tends to call for mathematician levels of mathematical rigor, while most of physics does not call for Real Analysis class mathematical rigor.
  4. May 4, 2005 #3
    It's only nonstandard in the US educational system. It's not an uncommon designation for a course at an Ontario university (high school courses are not nearly so specialized until the senior year).
  5. May 4, 2005 #4
    I am just a padawan myself, singleton, but your post interests me anyhow because I am doing a similar thing in my matriculation.
    Further down the track I believe I may do a graduate study in physics and I am trying also to base my choices in my math major somewhat on preparing for that.

    Applying some of my own logic to the information you give I suspect that the key difference between the two courses in algebra that you are offered is your own initiative, something like ohwilleke has said, which kind of physicist you would like to be. Algebra I looks like a generalised open ended study in the principles and concepts of math theory without solidifying the applications, while the Linear Algebra I seems to offer in contrast a solidifying experience in the applications of the theory. What I would do is to take a look at the particular subjects of physics that seem to cover the ground you feel you want to, and discover what prerequisite study is involved in matriculating there. In theory the two subjects look as though they are almost the same and will lend to the same fields of study in physics, with perhaps the one exception that if you have a specialist area of physics in mind that you would like to practice then Linear Algebra I could be the only way there. In any case you should check out the physics offerings to be sure..
    I hope that is clear and helpful.. May the force be with you..
  6. May 4, 2005 #5
    At the moment I am interested in:
    - GR
    - QM
    - high energy theory
    - astrophysics
    - the use of physics in engineering tasks

    I've learned a little about string theory (a few readings and the NOVA video by Greene) but not enough to know whether or not it interests me :P

    Basically I am as undecided as they come! :rofl: :tongue2:
    Last edited: May 4, 2005
  7. May 4, 2005 #6


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    maybe the difference is that the first course (lin alg) emphasizes matrcies, and the other emphasizes linear maps. for, physics i recommend the linear maps version as it is coordinate free, and physics is about coordinate free phenomena.
  8. May 4, 2005 #7


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    It doesn't look like Algebra I would rule out any options for you, making that supposedly the better choice (and probably the one with the sharper students). But, studying Martixes without an emphasis on applications would drive me bats (and I was a math major).
  9. May 5, 2005 #8
    I don't have too much to add. It looks like people have summarized the differences the same way I would have. I just wanted to say (from the perspective of a math grad student whose undergrad wasn't that long ago - also a fellow Canadian) is don't sweat these decisions too much. It sounds like you are pretty astute and definitely interested in what your studying. By the time you are done the program you will probably need to know the material covered in both courses. I'm not saying you need to take both courses; I'm saying if you are the kind of student who prepares for your courses the summer before the courses then you will have no problem picking up the material from the other class as you go.

    All I'm saying is it's probably fine to base your decision on what fits into your schedule better or which class is smaller. I promise you that there is no 'wrong' decision at this point. Not picking the right one will not stear the rest of your career into any particular direction. Right now I'm supposedly an Algebraic Number Theorist but I only took a single class in number theory as an undergrad. But the prof who taught my analysis class invited me to do a graduate degree with him and he's a number theorist so now that's what I'm working in. Your classes should reflect your interests but they do not dictate your interests.

    Hope some of that is helpful,
  10. May 5, 2005 #9
    at UFT? hmm i believe its a numerical/practical vs theory differences...but it could be a smart vs dumb difference. Either way if your looking to do physics(not eng.phys)

    take the theoretical/smart class. Check the engineering sections to see which course they require OR the social science...engineering usually takes the practical course and the social sciences take the dumb one....so take the other.
    Hope that helps
    oh yeah you'll most likely want to take the ALG I based on the description...
    Last edited: May 5, 2005
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