Results 1 to 4 of 4

Thread: find the n transformation

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    6

    find the n transformation

    let $\displaystyle T\in\mathcal L(\mathbb R^3),$ such that $\displaystyle T(x,y,z)=(ux+y,uy+z,uz).$ Compute $\displaystyle T^n(x,y,z).$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14
    obviously for each $\displaystyle 0\ne u\in\mathbb R.$

    as for the problem, compute the matrix associated to the transformation (it's the canonical basis for $\displaystyle \mathbb R^3,$ so it's easy) and then compute the characteristic polynomial.

    use the division algorithm and put $\displaystyle t^n=(u-\lambda)^3f(t)+at^2+bt+c,$ (1) where $\displaystyle (u-\lambda)^3$ is the characteristic polynomial and $\displaystyle at^2+bt+c$ is the remainder (its degree is one less than the characteristic polynomial), now you gotta compute $\displaystyle a,b,c.$

    first put $\displaystyle t=u,$ and differentiate once and evaluate again at $\displaystyle t=u,$ differentiate again and evaluate at $\displaystyle t=u,$ this will be useful to find $\displaystyle a,b,c.$

    once got those values, put them at (1) and then evaluate the associated matrix to the transformation; by the Cayley - Hamilton theorem, the matrix evaluated in the characteristic polynomial produces the null one so you'll end up (say the matrix is $\displaystyle A$) $\displaystyle A^n=aA^2+bA+cI_3.$

    finally, explicitly find $\displaystyle A^n$ and multiply it by $\displaystyle \left[ \begin{matrix}
    x \\
    y \\
    z
    \end{matrix} \right],$ and we're done!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Morgan View Post
    let $\displaystyle T\in\mathcal L(\mathbb R^3),$ such that $\displaystyle T(x,y,z)=(ux+y,uy+z,uz).$ Compute $\displaystyle T^n(x,y,z).$
    a very easy induction on $\displaystyle n$ will prove that: $\displaystyle T^n(x,y,z)=(u^nx + nu^{n-1}y+ \frac{n(n-1)}{2}u^{n-2}z, \ u^ny + nu^{n-1}z, \ u^n z).$
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,781
    Thanks
    3030
    Quote Originally Posted by Morgan View Post
    let $\displaystyle T\in\mathcal L(\mathbb R^3),$ such that $\displaystyle T(x,y,z)=(ux+y,uy+z,uz).$ Compute $\displaystyle T^n(x,y,z).$
    Write T as a matrix and calculate the first few powers of T. You should see the pattern pretty quickly.

    Or you could just calculate them directly from that formula, but I think it is easier to see with the matrix.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the laplace transformation.
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: Dec 12th 2011, 02:55 PM
  2. Find Möbius transformation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 4th 2011, 04:52 AM
  3. Find if a linear transformation is flattening
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 31st 2010, 08:21 AM
  4. Find a Mobius transformation...
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Nov 1st 2009, 07:30 AM
  5. Find a linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Feb 1st 2008, 10:34 AM

Search Tags


/mathhelpforum @mathhelpforum