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Thread: matrix representing L

  1. #1
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    matrix representing L

    Let L: $\displaystyle R^{2}->R^{2}$ be defined by $\displaystyle L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix }x+2y\\2x-y\end{pmatrix}$.
    Let S be the natural basis for $\displaystyle R^{2}$ and let $\displaystyle T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix} }$ be another basus for $\displaystyle R^{2}$. Find the matrix representing L with respect to:
    a) S
    b) S and T
    c) T and S
    d) T
    e) Compute L$\displaystyle (\begin{pmatrix}1\\2\end{pmatrix})$ 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.
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  2. #2
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    Quote Originally Posted by antman View Post
    Let L: $\displaystyle R^{2}->R^{2}$ be defined by $\displaystyle L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix }x+2y\\2x-y\end{pmatrix}$.
    Let S be the natural basis for $\displaystyle R^{2}$ and let $\displaystyle T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix} }$ be another basus for $\displaystyle R^{2}$. Find the matrix representing L with respect to:
    a) S
    b) S and T
    c) T and S
    d) T
    e) Compute L$\displaystyle (\begin{pmatrix}1\\2\end{pmatrix})$ 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 $\displaystyle L(0,1),L(1,0)$ as a coloumn vector and construct the matrix $\displaystyle [ L(0,1) | L(1,0) ]$.

    To find the matrix with respect to $\displaystyle T$ compute $\displaystyle [ L(-1,2)]_T, [L(2,0)]_T$ where $\displaystyle [ ~ ~ ]_T$ means you compute the coordinate of that coloum vector with respect to $\displaystyle T$. Then form the matrix of those coloumn vectors as you done so above.
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  3. #3
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    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 $\displaystyle \begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\ \0\end{pmatrix}$? I am really clueless.
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  4. #4
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    Quote Originally Posted by antman View Post
    S is the standard basis that equals $\displaystyle \begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\ \0\end{pmatrix}$? I am really clueless.
    Almost, you just go it backwards.

    The standard basis for $\displaystyle \mathbb{R}^2$ is $\displaystyle (1,0),(0,1)$ (in that order). The standard basis for $\displaystyle \mathbb{R}^3$ is (if you taken Calculus 3 you seen this before) $\displaystyle \bold{i},\bold{j},\bold{k}$ where $\displaystyle \bold{i}=(1,0,0), \bold{j}=(0,1,0), \bold{k}=(0,0,1)$. In general for $\displaystyle \mathbb{R}^n$ define $\displaystyle \bold{e}_j = (0,0,...,1,0,...,0)$ (here $\displaystyle 1\leq j\leq n$) where $\displaystyle 1$ is in the $\displaystyle j$-th position, then the standard basis is $\displaystyle \bold{e}_1,\bold{e}_2,...,\bold{e}_n$. It is easy to express everything in terms of a standard basis because $\displaystyle (a_1,...,a_n) = a_1\bold{e}_1 + ... + a_n \bold{e}_n$.
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  5. #5
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    For the matrix representing L with respect to S, I did $\displaystyle L(v_{1})={(1+2(0)),(2(1)-0)}=(1,2)$ and $\displaystyle L(v_{2})=(2,1)$ to get $\displaystyle \begin{pmatrix}1&2\\2&-1\end{pmatrix}$.

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

    For S and T, I got the matrix $\displaystyle \begin{pmatrix}3&-4\\2&4\end{pmatrix}$ and $\displaystyle \begin{pmatrix}3&2\\-4&4\end{pmatrix}$ 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 10:16 PM.
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