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Math Help - 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: \Re^2 --> \Re^3 be defined by

    T \begin{pmatrix}<br />
x_1\\<br />
x_2<br /> <br />
\end{pmatrix}<br /> <br />
= \begin{pmatrix}<br />
x_1 - x_2\\<br />
x_1 + 2x_2\\<br />
2x_1 - x_2<br />
\end{pmatrix}

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

    where u_1 = \begin{pmatrix}<br />
1\\<br />
2<br />
\end{pmatrix}

    u_2 = \begin{pmatrix}<br />
-1\\<br />
1<br />
\end{pmatrix}

    and
    v_1 = \begin{pmatrix}<br />
 1\\<br />
 0\\<br />
1<br />
 \end{pmatrix}, v_2 = \begin{pmatrix}<br />
  0\\<br />
  1\\<br />
1<br />
  \end{pmatrix}, v_3 = \begin{pmatrix}<br />
  1\\<br />
  1\\<br />
0<br />
  \end{pmatrix}

    I know that

    T (u_1) = \begin{pmatrix}<br />
 -1\\<br />
  5\\<br />
0<br />
  \end{pmatrix} and

    T (u_2) = \begin{pmatrix}<br />
 -2\\<br />
  1\\<br />
-3<br />
  \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: \Re^2 --> \Re^3 be defined by

    T \begin{pmatrix}<br />
x_1\\<br />
x_2<br /> <br />
\end{pmatrix}<br /> <br />
= \begin{pmatrix}<br />
x_1 - x_2\\<br />
x_1 + 2x_2\\<br />
2x_1 - x_2<br />
\end{pmatrix}

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

    where u_1 = \begin{pmatrix}<br />
1\\<br />
2<br />
\end{pmatrix}

    u_2 = \begin{pmatrix}<br />
-1\\<br />
1<br />
\end{pmatrix}

    and
    v_1 = \begin{pmatrix}<br />
1\\<br />
0\\<br />
1<br />
\end{pmatrix}, v_2 = \begin{pmatrix}<br />
0\\<br />
1\\<br />
1<br />
\end{pmatrix}, v_3 = \begin{pmatrix}<br />
1\\<br />
1\\<br />
0<br />
\end{pmatrix}

    I know that

    T (u_1) = \begin{pmatrix}<br />
-1\\<br />
5\\<br />
0<br />
\end{pmatrix} and

    T (u_2) = \begin{pmatrix}<br />
-2\\<br />
1\\<br />
-3<br />
\end{pmatrix}

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

    therefore the matrix you're looking for is: [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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