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Thread: Find matrix for T for bases B and B'

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Find matrix for T for bases B and B'

    Let T:$\displaystyle \Re^2 --> \Re^3$ be defined by

    T$\displaystyle \begin{pmatrix}
    x_1\\
    x_2

    \end{pmatrix}

    $ = $\displaystyle \begin{pmatrix}
    x_1 - x_2\\
    x_1 + 2x_2\\
    2x_1 - x_2
    \end{pmatrix}$

    Find the matrix for T with respect to the bases B = {$\displaystyle u_1, u_2$} and B' = {$\displaystyle v_1, v_2, v_3$}

    where $\displaystyle u_1 = \begin{pmatrix}
    1\\
    2
    \end{pmatrix}$

    $\displaystyle u_2 = \begin{pmatrix}
    -1\\
    1
    \end{pmatrix}$

    and
    $\displaystyle v_1 = \begin{pmatrix}
    1\\
    0\\
    1
    \end{pmatrix}$, $\displaystyle v_2 = \begin{pmatrix}
    0\\
    1\\
    1
    \end{pmatrix}$, $\displaystyle v_3 = \begin{pmatrix}
    1\\
    1\\
    0
    \end{pmatrix}$

    I know that

    T$\displaystyle (u_1) = \begin{pmatrix}
    -1\\
    5\\
    0
    \end{pmatrix}$ and

    T$\displaystyle (u_2) = \begin{pmatrix}
    -2\\
    1\\
    -3
    \end{pmatrix}$

    but i dont know how to do the rest.
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  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by flaming View Post
    Let T:$\displaystyle \Re^2 --> \Re^3$ be defined by

    T$\displaystyle \begin{pmatrix}
    x_1\\
    x_2

    \end{pmatrix}

    $ = $\displaystyle \begin{pmatrix}
    x_1 - x_2\\
    x_1 + 2x_2\\
    2x_1 - x_2
    \end{pmatrix}$

    Find the matrix for T with respect to the bases B = {$\displaystyle u_1, u_2$} and B' = {$\displaystyle v_1, v_2, v_3$}

    where $\displaystyle u_1 = \begin{pmatrix}
    1\\
    2
    \end{pmatrix}$

    $\displaystyle u_2 = \begin{pmatrix}
    -1\\
    1
    \end{pmatrix}$

    and
    $\displaystyle v_1 = \begin{pmatrix}
    1\\
    0\\
    1
    \end{pmatrix}$, $\displaystyle v_2 = \begin{pmatrix}
    0\\
    1\\
    1
    \end{pmatrix}$, $\displaystyle v_3 = \begin{pmatrix}
    1\\
    1\\
    0
    \end{pmatrix}$

    I know that

    T$\displaystyle (u_1) = \begin{pmatrix}
    -1\\
    5\\
    0
    \end{pmatrix}$ and

    T$\displaystyle (u_2) = \begin{pmatrix}
    -2\\
    1\\
    -3
    \end{pmatrix}$

    but i dont know how to do the rest.
    next you need to write $\displaystyle T(u_1), T(u_2)$ as a linear combination of $\displaystyle v_1,v_2,v_3.$ that is, $\displaystyle T(u_1)=-3v_1+3v_2+2v_3$ and $\displaystyle T(u_2)=-3v_1+v_3.$ so $\displaystyle [T(u_1)]_{B'}=\begin{pmatrix}-3 \\ 3 \\ 2 \end{pmatrix}$ and $\displaystyle [T(u_2)]_{B'}=\begin{pmatrix}-3 \\ 0 \\ 1 \end{pmatrix}.$

    therefore the matrix you're looking for is: $\displaystyle [T]_B^{B'}=\begin{pmatrix}[T(u_1)]_{B'} & [T(u_2)]_{B'} \end{pmatrix}=\begin{pmatrix}-3 & -3 \\ 3 & 0 \\ 2 & 1 \end{pmatrix}.$
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