Results 1 to 5 of 5

Math Help - matrix representing L

  1. #1
    Banned
    Joined
    Feb 2009
    Posts
    97

    matrix representing L

    Let L: R^{2}->R^{2} be defined by L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix  }x+2y\\2x-y\end{pmatrix}.
    Let S be the natural basis for R^{2} and let T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix}  } be another basus for R^{2}. Find the matrix representing L with respect to:
    a) S
    b) S and T
    c) T and S
    d) T
    e) Compute L (\begin{pmatrix}1\\2\end{pmatrix}) using the definition of L and also using the matrices obtained in (a), (b), (c) and (d).

    Any help would be incredibly appreciated or an example of how a problenm like this is to be done.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by antman View Post
    Let L: R^{2}->R^{2} be defined by L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix  }x+2y\\2x-y\end{pmatrix}.
    Let S be the natural basis for R^{2} and let T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix}  } be another basus for R^{2}. Find the matrix representing L with respect to:
    a) S
    b) S and T
    c) T and S
    d) T
    e) Compute L (\begin{pmatrix}1\\2\end{pmatrix}) using the definition of L and also using the matrices obtained in (a), (b), (c) and (d).

    Any help would be incredibly appreciated or an example of how a problenm like this is to be done.
    To find the matrix to the standard basis compute L(0,1),L(1,0) as a coloumn vector and construct the matrix [ L(0,1) | L(1,0) ].

    To find the matrix with respect to T compute [ L(-1,2)]_T, [L(2,0)]_T where [ ~ ~ ]_T means you compute the coordinate of that coloum vector with respect to T. Then form the matrix of those coloumn vectors as you done so above.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Feb 2009
    Posts
    97
    This question is still not making sense to me. I apologize. I'm trying to learn way too much in such a short amount of time. S is the standard basis that equals \begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\  \0\end{pmatrix}? I am really clueless.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by antman View Post
    S is the standard basis that equals \begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\  \0\end{pmatrix}? I am really clueless.
    Almost, you just go it backwards.

    The standard basis for \mathbb{R}^2 is (1,0),(0,1) (in that order). The standard basis for \mathbb{R}^3 is (if you taken Calculus 3 you seen this before) \bold{i},\bold{j},\bold{k} where \bold{i}=(1,0,0), \bold{j}=(0,1,0), \bold{k}=(0,0,1). In general for \mathbb{R}^n define \bold{e}_j = (0,0,...,1,0,...,0) (here 1\leq j\leq n) where 1 is in the j-th position, then the standard basis is \bold{e}_1,\bold{e}_2,...,\bold{e}_n. It is easy to express everything in terms of a standard basis because (a_1,...,a_n) = a_1\bold{e}_1 + ... + a_n \bold{e}_n.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Feb 2009
    Posts
    97
    For the matrix representing L with respect to S, I did L(v_{1})={(1+2(0)),(2(1)-0)}=(1,2) and L(v_{2})=(2,1) to get \begin{pmatrix}1&2\\2&-1\end{pmatrix}.

    Using this process for s, to find matrix with respect to T, gave me (3,-4) and (2,4) that i put into matrix \begin{pmatrix}-1&2&3&2\\2&0&-4&4\end{pmatrix} that i reduced and took right side to get matrix with respect to T=\begin{pmatrix}-2&2\\1/2&2\end{pmatrix}.

    For S and T, I got the matrix \begin{pmatrix}3&-4\\2&4\end{pmatrix} and \begin{pmatrix}3&2\\-4&4\end{pmatrix} for T and S.

    I am more confident in my answers to part a (S) and part d (T) because I really don't know how to do the others. Can someone help point me in the right direction?
    Last edited by antman; May 1st 2009 at 11:16 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Representing DE as systems
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: January 16th 2012, 11:23 AM
  2. Replies: 1
    Last Post: December 8th 2010, 08:07 AM
  3. Representing two's compliment
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 14th 2010, 09:41 AM
  4. Help for representing sets
    Posted in the Discrete Math Forum
    Replies: 14
    Last Post: January 9th 2010, 10:51 AM
  5. 2 x 2 matrix representing a rotation
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 23rd 2009, 06:59 AM

Search Tags


/mathhelpforum @mathhelpforum