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Thread: Change of Basis.

  1. #1
    Junior Member pearlyc's Avatar
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    Change of Basis.

    Okay, I did most of the parts in this question, just wanted you guys to check if it's the correct answer and teach me on the parts that I don't know how to do! Thanks a lot guys.

    Question :
    Let S denote the standard basis for $\displaystyle \mathbb{R}^2$ and B = { $\displaystyle [\begin{matrix} 1 \\ 2\end{matrix}]$ , $\displaystyle [\begin{matrix} -1 \\ 0\end{matrix}]$ } be another basis.

    (a) write down the change of basis matrix P B->S, from the basis B to the basis S.

    P B->S = $\displaystyle [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$

    (b) Hence find the change of basis matrix P S->B, from the basis S to the basis B.

    P S-> B = inverse of (P B->S)
    = inverse of $\displaystyle [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$
    = 1/2 $\displaystyle [\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]$

    (c) Find $\displaystyle [x]_B$ if x = (4,-1)

    I don't know how to do this one, can you guys help me out?

    (d) Show that $\displaystyle T : \mathbb{R}^2 -> \mathbb{R}^2$ defined by :
    $\displaystyle T(x,y) = (-x, + 2y, 3y)$

    Let x = [ x1, x2], y = y1, y2]

    T(x+y)= T(x1 + y1, x2 + y2) = (-x1 + 2y1, 3y1)
    = ( -x1, 0.x2) + (2y1, 3y2)
    = T(x1,x2) + T(y1,y2)

    T(ax) = T(ax1, ax2)
    = (-ax1, a.0)
    = a (-x1, 0)
    = aT(x)


    (e) Find the matrix representation of T with respect to the standard basis $\displaystyle [T]_s$

    $\displaystyle A_T$ = $\displaystyle [\begin{matrix} -1 & 2 \\ 0 & 3\end{matrix}]$


    (f) Use your answers from above to find the matrix representation of T with respect to the basis B, $\displaystyle [T]_B$

    $\displaystyle T[b] = P S->B . [T]_S . P B-> S$
    = $\displaystyle [\begin{matrix} 3 & 0 \\ 0 & -1\end{matrix}]$


    (g) Find $\displaystyle [T(x)]_B$ where $\displaystyle [x]_B$ = $\displaystyle [\begin{matrix} -3 \\ 5\end{matrix}]$

    $\displaystyle [Tx]_B$ = [tex][T]_B[x]_B = $\displaystyle [\begin{matrix} -9 \\ 5\end{matrix}]$
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  2. #2
    MHF Contributor Reckoner's Avatar
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    Hi, pearlyc!

    Quote Originally Posted by pearlyc View Post
    Let S denote the standard basis for $\displaystyle \mathbb{R}^2$ and B = { $\displaystyle [\begin{matrix} 1 \\ 2\end{matrix}]$ , $\displaystyle [\begin{matrix} -1 \\ 0\end{matrix}]$ } be another basis.

    (a) write down the change of basis matrix P B->S, from the basis B to the basis S.

    P B->S = $\displaystyle [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$
    Correct!

    Quote Originally Posted by pearlyc View Post
    (b) Hence find the change of basis matrix P S->B, from the basis S to the basis B.

    P S-> B = inverse of (P B->S)
    = inverse of $\displaystyle [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$
    = 1/2 $\displaystyle [\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]$
    Correct!

    Quote Originally Posted by pearlyc View Post
    (c) Find $\displaystyle [x]_B$ if x = (4,-1)

    I don't know how to do this one, can you guys help me out?
    What's the trouble? You have a vector with coordinates relative to the standard basis, and need to find the coordinates relative to the basis $\displaystyle B$. Use your transition matrix!

    Quote Originally Posted by pearlyc View Post
    (d) Show that $\displaystyle T : \mathbb{R}^2 -> \mathbb{R}^2$ defined by :
    $\displaystyle T(x,y) = (-x, + 2y, 3y)$

    Let x = [ x1, x2], y = y1, y2]

    T(x+y)= T(x1 + y1, x2 + y2) = (-x1 + 2y1, 3y1)
    = ( -x1, 0.x2) + (2y1, 3y2)
    = T(x1,x2) + T(y1,y2)

    T(ax) = T(ax1, ax2)
    = (-ax1, a.0)
    = a (-x1, 0)
    = aT(x)

    Are you being asked to prove that $\displaystyle T$ is a linear transformation?

    Your work is completely wrong. I have highlighted your errors in red.

    Follow the definition of $\displaystyle T$, and don't confuse the different sets of coordinates that you have in use. For example, here is how you could do the first part:

    $\displaystyle T(x,\ y) = (-x + 2y,\ 3y)$

    $\displaystyle \text{Let }\textbf{u} = (u_1,\ u_2)\text{ and }\textbf{v} = (v_1,\ v_2)$

    $\displaystyle \text{So, }T(\textbf{u} + \textbf{v})$

    $\displaystyle =T(u_1 + v_1,\ u_2 + v_2)$

    $\displaystyle =\left(
    -(u_1 + v_1) + 2(u_2 + v_2),\ 3(u_2 + v_2)
    \right)$

    $\displaystyle =\left(
    -u_1 - v_1 + 2u_2 + 2v_2,\ 3u_2 + 3v_2
    \right)$

    $\displaystyle =\left(
    -u_1 + 2u_2 - v_1 + 2v_2,\ 3u_2 + 3v_2
    \right)$

    $\displaystyle =
    (-u_1 + 2u_2,\ 3u_2) + (-v_1 + 2v_2,\ 3v_2)
    $

    $\displaystyle =
    T(u_1,\ u_2) + T(v_1,\ v_2)
    $

    $\displaystyle =
    T(\textbf{u}) + T(\textbf{v})
    $

    See if you can prove the second property now.

    Quote Originally Posted by pearlyc View Post
    (e) Find the matrix representation of T with respect to the standard basis $\displaystyle [T]_s$

    $\displaystyle A_T$ = $\displaystyle [\begin{matrix} -1 & 2 \\ 0 & 3\end{matrix}]$
    Correct!

    Quote Originally Posted by pearlyc View Post
    (f) Use your answers from above to find the matrix representation of T with respect to the basis B, $\displaystyle [T]_B$

    $\displaystyle T[b] = P S->B . [T]_S . P B-> S$
    = $\displaystyle [\begin{matrix} 3 & 0 \\ 0 & -1\end{matrix}]$
    Looks good to me.

    Quote Originally Posted by pearlyc View Post
    (g) Find $\displaystyle [T(x)]_B$ where $\displaystyle [x]_B$ = $\displaystyle [\begin{matrix} -3 \\ 5\end{matrix}]$

    $\displaystyle [Tx]_B$ = [tex][T]_B[x]_B = $\displaystyle [\begin{matrix} -9 \\ 5\end{matrix}]$
    Check your multiplication here.
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  3. #3
    Junior Member pearlyc's Avatar
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    Thanks for your help and effort to check!

    I still don't know how to approach part (c) though, haha.
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  4. #4
    Moo
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    Hello,

    Quote Originally Posted by pearlyc View Post
    Thanks for your help and effort to check!

    I still don't know how to approach part (c) though, haha.
    If P is the transition matrix between basis E and F, then vector X(x1, x2) in E will have the following coordinates in F : PX
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