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Math Help - Linear Algebra Question

  1. #1
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    Linear Algebra Question

    I'm completly unsure how to tackle this problem

    the subject of the following is Linear Transformations.

    Given in the text:

    ....We also have linear transformations on function spaces, such as multiplication operators.
    2.5
    <br />
  M_f :C^k (I) \to C^k (I),M_f g(x) = f(x)g(x) <br />

    given that:
    f \in C^k (I),I = [a,b] and the operation of differentiation

    2.6
    <br />
  D:C^{k + 1} (I) \to C^k (I),Df(x) = f'(x) <br />

    we also have integration

    2.7
    <br />
  i:C^k (I) \to C^{k + 1} (I),if(x) = \int\limits_a^x {f(y)dy} <br />

    note also that
    2.8
    <br />
  D:P_{k + 1}  \to P_k<br />

    and

    <br />
  i:P_k  \to P_{k + 1}<br />

    where  P_k denotes the space of polynomials in x of degree  \leqslant k




    QUESTION:

    compute
     Di and  iD

    and specify the null space and range of both i and D
    Last edited by cliggax; March 15th 2008 at 05:40 PM. Reason: To clarify
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  2. #2
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    It is completely understandable that you may not know that the greatest problem in mathematics communication is the lack of a standard in notation. I dare say that only the authors of your text and their fellows can read that notation. So if you want help, you must explain what those terms and symbols mean.
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  3. #3
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    I have edited my question as best as possible. I am still lost on this question. any advice would be of help.
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  4. #4
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    The polynomial space of degree k has k+1 basis vectors.

    to construct a matrix transformation we Transform each of the basis.

    See the below attachments.

    I did an example D the derivative transformation on p_2

    note D goes from P k+1 to P k and

    i goes from P k to p k+1

    Linear Algebra Question-capture.jpg

    Linear Algebra Question-capture2.jpg

    Linear Algebra Question-capture3.jpg
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  5. #5
    Behold, the power of SARDINES!
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    Cont.

    Now you would need to do the same thing for

    i the integral transform,

    Then compute the product of the two matricies.

    I hope this helps.
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  6. #6
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    is the null space of the D transformation you calculated any point on the z axis? and is the range the xy plane?
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  7. #7
    Behold, the power of SARDINES!
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    Quote Originally Posted by cliggax View Post
    is the null space of the D transformation you calculated any point on the z axis? and is the range the xy plane?
    The null space of the D transform would be any element of the form

    a \cdot e_1 +0\cdot e_2 +....0 \cdot e_{k+1}

    The null space is the set of constant polynomials. It is everything that is transformed into the Zero Vector

    The range would be the span of the column Vectors

    The column vectors span the subspace P_k of P_{k+1}
    Last edited by TheEmptySet; March 15th 2008 at 06:48 PM. Reason: To be clearer
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  8. #8
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    is the function f(x) = 2x^2 +3x - 1 an arbitrary function? Essentially by multiplying

    -1
    3
    2

    by the other matrix are you finding the D transform of the function 2x^2 +3x - 1?

    Could i substitute my own function in place of yours (of degree 2 of course)
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  9. #9
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Yes for example

    f(x)=22x^2-8x+7 could be written as

     7 \cdot e_1 -8 \cdot e_2+ 22 \cdot e_3 or in vector form

    7 \cdot \begin{bmatrix}<br />
1 \\<br />
0 \\<br />
0<br />
\end{bmatrix}<br />
-8 \cdot \begin{bmatrix}<br />
0 \\<br />
1 \\<br />
0<br />
\end{bmatrix}<br />
+ 22 \cdot <br />
\begin{bmatrix}<br />
0 \\<br />
0 \\<br />
1<br />
\end{bmatrix}<br />
=<br />
\begin{bmatrix}<br />
7 \\<br />
-8 \\<br />
22<br />
\end{bmatrix}<br />
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