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Vector Spaces Vocab

  1. Mar 7, 2003 #1
    I have an upcoming Linear Algebra exam and my textbooks are really vague in defining certain concepts (and he didn't limit the ambiguous nature to Linear Algebra. His Calculus book is the same way).

    Would someone mind helping me define or determine Determination tests for concepts like span, subspace, basis, rank, dimension, row space, column space, linear combination, etc.?

  2. jcsd
  3. Mar 8, 2003 #2


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    I think it will be better if you post a couple specific questions.
  4. Mar 8, 2003 #3
    Okay. It's really just a request for less vague definitions.

    I figured out the concept of span (but it took a while).

    However as for a specific question, how does one determine the row space and column space?
  5. Mar 16, 2003 #4
    You can find rigorous definitions at http://mathworld.wolfram.com (but there are often multiple equivalent ways to define something, so these may slightly differ from your book's.)

    IMHO you should start by reading and figuring out the exact definitions of a vector space and of a linear transform. Once you understand those, the other defs are easy.
  6. Mar 17, 2003 #5
    Well, the test was last Wednesday but I found the Schaum's Outlines for Linear Algebra and around four really good textbooks so I purchased them and studied throughout the break.

    I basically understand the concepts (the nature or neccessity is another story) so about the exam, I think I did decent (but I heard there was a curve so I just might get an A or a B (I'm sure it's at least a B without the curve).

    Thanks Damgo
  7. Mar 17, 2003 #6
    Hehe I'm currently learning very similar stuff (all you mentioned in first post) and am also yet to see either nature or necessity :smile:
  8. Mar 17, 2003 #7
    lol, I don't think there is any practical applications of it.

    Mathematicians just sat together one day to smoke some pot and BANG! Linear Algebra is born....
  9. Mar 17, 2003 #8
    Linear Algebra -> matrices -> tensors -> SR,GR,deformations,fluids....physics...
  10. Mar 17, 2003 #9


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    Electronics... control systems... computer graphics... deformations analysis...
  11. Mar 17, 2003 #10
    3-D video games and computer games use a lot of linear algebra. Bucketloads of it. So much, that game designers have developed or applied advanced compact techniques of representing linear transforms.
  12. Mar 17, 2003 #11
    My first "moment of clarity" in linalg was when I saw why matrices and linear transforms were the same thing.... work on figuring this out. :)

    Real 3-D space -- R3 -- is a happy vector space, and lots of the things we want to with, like rotations or scalings, are all linear transforms. Computers do incredibly large amounts of matrix manipulations, all the time.

    Plus about all of physics and lots of math is built on top of linear algebra... partially because as the old saw goes "everything is linear to first order." :) It seems weird at first, but its importance will soon become apparent... I hardly even think about it as math anymore because it's such a built-in part of the way I think....
  13. Mar 17, 2003 #12

    Tom Mattson

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    I took linear algebra my freshman year, and did not use much of it until my first year of grad school, in Quantum Mechanics I. Others were in a similar boat, and the professor found that he had to give us a refresher on the vector space axioms.
  14. Mar 17, 2003 #13
    Guys, I was joking when I had mentioned the lack of practical applications in Linear Algebra.

    However, I have yet to see an actual application of it since a good bit of it has been theoretical.

    Oh, before I forget, I got a 96 on my Linear Algebra exam! Second highest test score in the class so that's a bit of relief.

    Dr. Postell (my Linear Algebra professor): "...this is quite a common inner product. Physics majors <looks at me> will use this extensively..."

    Tom, would you mind showing us an example of how Linear Algebra comes into place with QM?

    Wow, that's cool. Want to hear something ridiculuous? Linear Algebra isn't a requirement for Physics majors at my school. In fact, I could drop the class and take a computer class and get my requirement filled for that area. I'm taking Linear Algebra out of concern because I know it'll come into place in Physics.
  15. Mar 17, 2003 #14
    what's linear algebra?
  16. Mar 17, 2003 #15
    Linear Algebra is the study of matrices and the properties of matrices.

    And Majinvegeta, I haven't forgot about my linear algebra notes to send to you. My Microsoft Word has developed a virus on it so sending notes is very difficult but I will send my notes as soon as I can.
  17. Mar 17, 2003 #16
    Hey Sting did you decide on which major you would take - Maths/phys or both I can't remember whether you did or not..
  18. Mar 17, 2003 #17
    Hey Mulder,

    Actually, Physics definitely, but I want to double major in either Physics and Mathematics or Physics and Electrical Engineering.

    The EE is for job security and finances but the math degree is for temporary job security, fun, and the potential opportunity for working in genetics (last time I read, they need statisticians).

    So, techinically, I'm still trying to look over all my options.

    Wish me luck! :smile:
  19. Mar 17, 2003 #18
    groovy! I can't wait.

    What is it used for? A lot of math(i'm thinking of fractals) doesn't apply to the physical world. So what's its use, how does it help us improve things?
  20. Mar 17, 2003 #19


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    The main importance, IMHO, is that Linear Algebra is a language, it gives us the descriptive power to talk about things that would otherwise be either very cumbersome or very vague.

    For instance, one of the things matrices give us is the power to talk about a system of equations as a single unit, and do a variety of operations that make sense to do on the system as a whole.

    Another thing it gives us is the power to speak rigorously about some geometric concepts. For example, consider the surface given by the parametric equations:

    x(s, t) = s
    y(s, t) = s^2 * t
    with -1 < s < 1 and -1 < t < 1

    If you plot this suface, you will notice that it's kinda sorta pinched into a 1-D surface at the origin. We can prove this rigorously as follows:

    First, compute the jacobian of the above equations. The jacobian of a system of functions is simply the matrix who's i-th row and j-th column contains the derivative of the i-th function with respect to the j-th variable. In this case, the jacobian of the transformation is:

    [1, 0]
    [2st, s^2]

    If you plug in 0 for s, you get:

    [1, 0]
    [0, 0]

    Which is a rank one matrix. (The rank of a matrix is the the dimension of its row space; i.e. the number of linearly independant row vectors in the matrix) Our vague idea of a 1-dimensional surface coincides exactly with jacobians that have rank 1! Similarly, if a system of equations describes a 2-d surface the matrix has rank 2. For example, at the point s = 0.5 t = 0.5 the jacobian of the above system is:

    [1, 0]
    [0.5, 0.25]

    Which is rank 2, as we'd expect from plotting the surface.


    P.S. bleh just saw where you said you're joking
    P.P.S can you do superscripts in 3.0?
    Last edited: Mar 17, 2003
  21. Mar 17, 2003 #20
    <gapes in disbelief> Congrats on your test tho!
    <smashes head repeatedly against the wall>

    Hurkyl hit the nail on the head.... trying to do QM without linear algebra is like trying to write an essay without sentences. For example, in the general formulation of QM:

    A state of a system is represented by a vector |[psi]> in a special type of complex vector space called a Hilbert space. Observable properties are represented by a special type of linear transformation called a Hermitian operator. (you will probably see symmetric matrices, which are very similar.)

    The average value of some observable A in state |[psi]> is then the inner product of A([psi]) and A, written <[psi]|A|[psi]>. And so on.... in general the inner product tells you how much "overlap" ther is between two states. It takes a while to grasp how it all fits together, but it's so pretty when it does...
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