Results 1 to 3 of 3

Thread: Two Linear Transformation Problems

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    14

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

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


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

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

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

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



    For 1), I found $\displaystyle L(1) = 6, L(x) = 19, and L(x^2)= 2x+72$. 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; Sep 14th 2011 at 08:18 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Mar 2011
    Posts
    40

    Re: Two Linear Transformation Problems

    for 2a):

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

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

    for 2b):

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

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


    for 2c):

    $\displaystyle L(E_{11}) = ME_{11}= $$\displaystyle \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 $\displaystyle L$ with respect to $\displaystyle E$ is $\displaystyle \left(\begin{array}{cc}3\\ 0\\ 5\\ 0\end{array}\right)$.

    In a similar way you can find the rest three columns.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    14

    Re: Two Linear Transformation Problems

    Thank you very much. I feel incredibly dense for not having seen, especially since I had been staring at $\displaystyle M = c_{1}E_{11} + c_{2}E_{12} + c_{3}E_{21} + c_{4}E_{22}$ 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 $\displaystyle P_{3}$, and then putting those in an augmented matrix with the ordered bases of $\displaystyle P_{2}$ (interpreted as (1, 0) and (0,1)) gives the matrix representation.

    Thanks very much for the help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Aug 1st 2011, 10:00 PM
  2. linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 28th 2009, 06:40 AM
  3. Linear Algebra.Linear Transformation.Help
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 5th 2009, 01:14 PM
  4. Linear Transformation Problems
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 3rd 2008, 07:04 PM
  5. Graph Transformation Problems
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 16th 2007, 06:01 PM

Search Tags


/mathhelpforum @mathhelpforum