Results 1 to 4 of 4

Math Help - find the n transformation

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    6

    find the n transformation

    let T\in\mathcal L(\mathbb R^3), such that T(x,y,z)=(ux+y,uy+z,uz). Compute 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
    3,654
    Thanks
    13
    obviously for each 0\ne u\in\mathbb R.

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

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

    first put t=u, and differentiate once and evaluate again at t=u, differentiate again and evaluate at t=u, this will be useful to find 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 A) A^n=aA^2+bA+cI_3.

    finally, explicitly find A^n and multiply it by \left[ \begin{matrix}<br />
   x  \\<br />
   y  \\<br />
   z  <br />
\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 T\in\mathcal L(\mathbb R^3), such that T(x,y,z)=(ux+y,uy+z,uz). Compute T^n(x,y,z).
    a very easy induction on n will prove that: 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
    15,525
    Thanks
    1384
    Quote Originally Posted by Morgan View Post
    let T\in\mathcal L(\mathbb R^3), such that T(x,y,z)=(ux+y,uy+z,uz). Compute 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: December 12th 2011, 02:55 PM
  2. Find Möbius transformation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 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: November 1st 2009, 07:30 AM
  5. Find a linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 1st 2008, 10:34 AM

Search Tags


/mathhelpforum @mathhelpforum