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Math Help - Two Linear Transformation Problems

  1. #1
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    Two Linear Transformation Problems

    I think I've got a rather glaring hole in my understanding of this material. I've read and reread the text, watched the MIT Open Courseware lectures, and beat my head against this material for quite a while now. Any help would be much appreciated. Without further ado...

    1) Define a linear transformation L: P_{3} \rightarrow P_{2} by L(f(x) = f'(x) + \int_0^6 \! f(x) \, \dif x . Consider ordered bases B = {1, x, x^2} and C = {1, x} for  P_{3} and P_{2} respectively.

    a) Find the representation matrix A of L with respect to the given bases B and C.
    b) Express [L(a_{0} + a_{1}x + a_{2}x^2)]_C in terms of a_{0}, a_{1}, a_{2}


    2) Let \mathbb{R}^{2 x 2} be the vector space of all 2 x 2 matrices (over \mathbb{R}. Consider
     E_{11} = \begin{array}{|cc|}1&0\\0&0\end{array},  E_{12} = \begin{array}{|cc|}0&1\\0&0\end{array},  E_{21} = \begin{array}{|cc|}0&0\\1&0\end{array},  E_{22} = \begin{array}{|cc|}0&0\\0&1\end{array}; and M = \begin{array}{|cc|}3&4\\5&6\end{array}.

    It is known that E = E_{11}, E_{12}, E_{21}, E_{22} is an ordered basis for \mathbb{R}^{2 x 2}. Define L: \mathbb{R}^{2 x 2} \rightarrow \mathbb{R}^{2 x 2} by L(x) = Mx for all x = \begin{array}{|cc|}x_11&x_12\\x_21&x_22\end{array} \in \mathbb{R}^{2 x 2}.

    Edit: Couldn't get the format right for that x matrix. Is should be x_{11}, x_{12}, x_{21}, x_{22}.

    a) Find [M]_E, the coordinate vector of M with respect to the ordered basis E.
    b) Is L a linear transformation?
    c) Find the representation matrix of L with respect to E, if L is a linear transformation.



    For 1), I found L(1) = 6, L(x) = 19, and L(x^2)= 2x+72. Then I tried to set up a matrix with the bases of C and those values. I interpreted the bases of C as the standard bases (1, 0), (0, 1). I really don't think that's right, so I'm kind of stuck there.

    For 2), I can't even make headway on the first part. I set up the problem like I would to find a coordinate vector, but it comes out as a 2 x 10 matrix. Trying to account for the zero columns still leaves me with a 2 x 6. I believe that this is a linear transformation, but as I said, I'm quite stumped.

    Any help would be very much appreciated.
    Last edited by DarrenM; September 14th 2011 at 08:18 AM.
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  2. #2
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    Re: Two Linear Transformation Problems

    for 2a):

    Since M = 3E_{11}+4E_{12}+5E_{21}+6E_{22},

    the coordinate vector of M with respect to the ordered basis E is [M]_E = (3,4,5,6).

    for 2b):

    L(x+y) = M(x+y) = Mx+My = L(x)+L(y),
    L(\lambda x) = M(\lambda x) = \lambda Mx = \lambda L(x),

    \forall \lambda \in \mathbb{R} and \forall x,y \in \mathbb R^{2x2} so  L is a linear transformation.


    for 2c):

    L(E_{11}) = ME_{11}= \left(\begin{array}{cc}3&4\\5&6\end{array}\right) \left(\begin{array}{cc}1&0\\0&0\end{array}\right) = \left(\begin{array}{cc}3&0\\5&0\end{array}\right)=  3E_{11}+0E_{12}+5E_{21}+0E_{22}

    so the first column of representation matrix of L with respect to E is \left(\begin{array}{cc}3\\ 0\\ 5\\ 0\end{array}\right).

    In a similar way you can find the rest three columns.
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  3. #3
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    Re: Two Linear Transformation Problems

    Thank you very much. I feel incredibly dense for not having seen, especially since I had been staring at  M = c_{1}E_{11} + c_{2}E_{12} + c_{3}E_{21} + c_{4}E_{22} for so long trying to make sense of a coordinate vector in terms of a 2 x 2 matrix.

    Once I read that comment the third part fell into place.

    As for problem 1, it turns out I was correct. Evaluating the linear transformation with the given ordered bases of P_{3}, and then putting those in an augmented matrix with the ordered bases of P_{2} (interpreted as (1, 0) and (0,1)) gives the matrix representation.

    Thanks very much for the help.
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