Results 1 to 3 of 3

Math Help - Linear Algebra help

  1. #1
    Junior Member
    Joined
    Apr 2008
    From
    Gainesville
    Posts
    68

    Unhappy Linear Algebra help

    Let T: R^2 --> R^3 be defined by T(a1,a2)=(a1-a2,a1,2a1+a2) Let beta be the standard basis for R^2 and gamma={(1,1,0),(0,1,1),(2,2,3)}.
    compute [T]_beta to gamma
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by JCIR View Post
    Let T: R^2 --> R^3 be defined by T(a1,a2)=(a1-a2,a1,2a1+a2) Let beta be the standard basis for R^2 and gamma={(1,1,0),(0,1,1),(2,2,3)}.
    compute [T]_beta to gamma
    "beta be the standard basis for R^2" => T(1,0) = (1,1,2) and T(0,1) = (-1,0,1)

    The question essentially asks:

    We want to write (a1-a2,a1,2a1+a2) as a linear combination of gamma. Let the coefficients of this linear combination be x,y,z in that order.

    Then given (a1,a2) in R^2, we want to find a matrix that maps it to T(a1,a2) which is written as a linear combination of gamma.

    Thus:

    T(a1,a2) = (a1-a2,a1,2a1+a2) = x(1,1,0)+y(0,1,1)+z(2,2,3)

    But observe that:

    (a1-a2,a1,2a1+a2) = \begin{pmatrix}1 & -1 \\ 1 & 0 \\ 2 & 1\end{pmatrix} \begin{pmatrix}a1\\ a2\end{pmatrix}

    x(1,1,0)+y(0,1,1)+z(2,2,3) = \begin{pmatrix}1 & 0 & 2 \\ 1 & 1 & 2 \\ 0 & 1 & 3\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}

    So:
    \begin{pmatrix}1 & -1 \\ 1 & 0 \\ 2 & 1\end{pmatrix} \begin{pmatrix}a1\\ a2\end{pmatrix} =  \begin{pmatrix}1 & 0 & 2 \\ 1 & 1 & 2 \\ 0 & 1 & 3\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}

    This means,

    \begin{pmatrix}1 & 0 & 2 \\ 1 & 1 & 2 \\ 0 & 1 & 3\end{pmatrix}^{-1}\begin{pmatrix}1 & -1 \\ 1 & 0 \\ 2 & 1\end{pmatrix} \begin{pmatrix}a1\\ a2\end{pmatrix} =  \begin{pmatrix}x\\ y\\ z\end{pmatrix}

    Thus the matrix \begin{pmatrix}1 & 0 & 2 \\ 1 & 1 & 2 \\ 0 & 1 & 3\end{pmatrix}^{-1}\begin{pmatrix}1 & -1 \\ 1 & 0 \\ 2 & 1\end{pmatrix} maps (a1,a2) to (x,y,z)


    I will let you compute that matrix....
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Isomorphism View Post
    The question essentially asks:

    We want to write (a1-a2,a1,2a1+a2) as a linear combination of gamma.
    thanks for that. for some reason, i do not recall the notation the OP used, so i was wondering what they were asking for
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 1st 2011, 11:00 PM
  2. Basic Linear Algebra - Linear Transformations Help
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 7th 2010, 04:59 PM
  3. Replies: 2
    Last Post: December 6th 2010, 04:03 PM
  4. Replies: 7
    Last Post: August 30th 2009, 11:03 AM
  5. Replies: 3
    Last Post: June 2nd 2007, 11:08 AM

Search Tags


/mathhelpforum @mathhelpforum