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Math Help - 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 \mathbb{R}^2 and B = { [\begin{matrix} 1 \\ 2\end{matrix}] , [\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 = [\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 [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]
    = 1/2 [\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]

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

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

    (d) Show that T : \mathbb{R}^2 -> \mathbb{R}^2 defined by :
    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 [T]_s

    A_T = [\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, [T]_B

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


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

    [Tx]_B = [tex][T]_B[x]_B = [\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 \mathbb{R}^2 and B = { [\begin{matrix} 1 \\ 2\end{matrix}] , [\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 = [\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 [\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]
    = 1/2 [\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]
    Correct!

    Quote Originally Posted by pearlyc View Post
    (c) Find [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 B. Use your transition matrix!

    Quote Originally Posted by pearlyc View Post
    (d) Show that T : \mathbb{R}^2 -> \mathbb{R}^2 defined by :
    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 T is a linear transformation?

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

    Follow the definition of 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:

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

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

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

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

    =\left(<br />
-(u_1 + v_1) + 2(u_2 + v_2),\ 3(u_2 + v_2)<br />
\right)

    =\left(<br />
-u_1 - v_1 + 2u_2 + 2v_2,\ 3u_2 + 3v_2<br />
\right)

    =\left(<br />
-u_1 + 2u_2 - v_1 + 2v_2,\ 3u_2 + 3v_2<br />
\right)

    =<br />
(-u_1 + 2u_2,\ 3u_2) + (-v_1 + 2v_2,\ 3v_2)<br />

    =<br />
T(u_1,\ u_2) + T(v_1,\ v_2)<br />

    =<br />
T(\textbf{u}) + T(\textbf{v})<br />

    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 [T]_s

    A_T = [\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, [T]_B

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

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

    [Tx]_B = [tex][T]_B[x]_B = [\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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