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Linear Algebra Problems

  1. Mar 17, 2006 #1
    I don't know if it is the subj or my prof :cry: but I am having a hard time grasping Linear Algebra. We are working on vector spaces - subspaces, basis & dimension, nullspace...etc. The text we have is LINEAR ALGEBRA WITH APPLICATIONS 7E by Steven J. Leon. I don't think there are enough problems for me to work through in the text. Does anyone know a web site or online text that has good problems and answers so I can get some practice in this stuff!!! Answers are critical and any thing that includes steps on how that answer was obtained would be awsome... ???? Or maybe there is somebody out there just itching to explain subspace & basis <<< mostly basis is where I am stuck right now >>> Thanks
     
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  3. Mar 17, 2006 #2

    mathwonk

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    the key idea is linear combinations. they are a way to, express an infinite number of things usaing only a finite number of basic ones.


    e.g. every polynomial of degree 2 or less can be expressed as a sum of multiples of (i.e. as a linear combination of) the absic monomials, 1,X,X^2. Hence the vector space of all polynomials of degree 2 or less has dimesion at most 3 (the number of elements in our set 1,X,X^2).

    the set 1,X,X^2 which can be used to express everything contain s a basis. To see if it is itself a basis, you must check whether any polynomial of degree 2 or less can be expressed using these guys, in more than one way.

    i.e. suppose a + bX + cX^2 = d + eX + fX^2. Can you be sure that a=d, b=e, and c = f? if so, then there is only one way to use the basic monomials 1,X,X^2 to express any degree 2 polynomial, and so they are a "basis". I.e. they not only "span" everything, but they are also independent, hence a basis.

    (My use of the word "basic" earlier was merely informal, and had nothing to do with the technical meaning of "basis".)


    the functions 1, cos^2 and sin^2, are not independent, since 1 = cos^2 + sin^2. so there is more than one way to express the function 1 using them.

    first master linear combinations, then an unnderstanding of bases will be added unto you.


    i.e. given a collection {v1,...vn}, the set V f all things that are linear combinations of these, i.e. all things of form a1v1+...anvn, is the subspace spanned by the set v1,...,vn.

    And if each thing in that subspace V has exactly one expression as a linear combination of the set v1,...vn, then that set is a basis of that subspace.

    in fact v1,...,vn is a basis of V if and only if no one of the vj can be expressed as a linear combination of the other v's.


    so all the concepts you are asking about are defiend in terms of linear combinations.
     
    Last edited: Mar 17, 2006
  4. Mar 17, 2006 #3

    matt grime

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    degree 2 polys have dimension 3.
     
  5. Mar 17, 2006 #4

    mathwonk

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    thanks matt, i went back and changed the 2 to a 3 where i noticed it.
     
  6. Mar 18, 2006 #5

    HallsofIvy

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    Now go back and change "dimension at most 3" to "dimension 3". You are, after all, talking about the vector space of all degree two polynomials.,
     
  7. Mar 18, 2006 #6
    thank you, mathwonk, for taking the time to explain that. It helps.
     
  8. Mar 19, 2006 #7

    mathwonk

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    thanks for the remark halls, but i don't want to change that statement, since at the time all i had done was prove my set was spanning, and not yet that it was a basis. so my argument showed not that the dimension was equal to 3, but only that it was at most 3.

    I never did finish the argument that it was a basis, so I was never entitled to say the dimension equalled 3. I left that as an exercise for the reader.

    but of course you are right, once one does prove that the set 1,X,X^2, is also independent, it follows that the space of polynomials of degree at most 2, is actually 3.
     
  9. Mar 19, 2006 #8

    mathwonk

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    one also needs to be careful about the definition of a polynomial. although with the standard definition, that two polynomials are equal if and only if their coefficients are all equal, the dimension of the space of polynomials of degree at most d is indeed d+1, some students confuse the concept of formal polynomial with that of polynomial function.


    A polynomial function is the image of a polynomial under the map from polynomials to functions. Over the field Z/2 e.g. X^2 = X, so as polynomial functions X^2 and X are equal. Then this space of functions defiend by polynomials of degree at most 2, actually has dimension 2.

    I.e. as functions 1,X,X^2 are not independent. so i omitted this part of the discussion as subtler than the easy spanning question.
     
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