Results 1 to 6 of 6

Thread: Interesting Linear Transformation question

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    175

    Interesting Linear Transformation question

    ***$\displaystyle U \text{ and } W \in R_4[x]$
    and below I simply transform from $\displaystyle R_4[x] \rightarrow R^4$

    $\displaystyle W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)$

    Does there exist a linear transformation $\displaystyle T:R_4[x] \rightarrow R_4[x]$ so that $\displaystyle T(U)=W$ and $\displaystyle T(W) = U$

    So I got to this...

    $\displaystyle W = Sp{(1,0,0,0),(0,1,0,-1)}$
    $\displaystyle U = Sp{(4,0,-1,0),(0,4,0,-1)} $

    And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

    I know in general that if you have a basis you can create any transformation you want but I don't know how to.

    Thanks!
    Last edited by jayshizwiz; Nov 3rd 2010 at 12:54 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2010
    Posts
    175
    Ok, I think I got it. It took me many pages on my notebook...

    Basically, $\displaystyle T(x,y,z,r) = (4x+15z, -5r-y, -x-4z, r)$

    Thanks anyway!

    ***and after that I must change it back to R_4[x]...
    Last edited by jayshizwiz; Nov 3rd 2010 at 12:57 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by jayshizwiz View Post
    $\displaystyle W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)$

    Does there exist a linear transformation $\displaystyle T:R_4[x] \rightarrow R_4[x]$ so that $\displaystyle T(U)=W$ and $\displaystyle T(W) = U$

    So I got to this...

    $\displaystyle W = Sp{(1,0,0,0),(0,1,0,-1)}$
    $\displaystyle U = Sp{(4,0,-1,0),(0,4,0,-1)} $

    And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

    I know in general that if you have a basis you can create any transformation you want but I don't know how to.

    Thanks!
    What exactly are these spaces?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2010
    Posts
    175
    Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

    I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by jayshizwiz View Post
    Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

    I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion
    Haha, the funny this is my problem was not knowing what $\displaystyle R_4[x]$ is? Maybe $\displaystyle \mathbb{R}_4[x]=\left\{a_0+\cdots+a_4x^4:a_0,\cdots,a_4\in\mathbb {R}\right\}$ but from the elements I would assume not.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2010
    Posts
    175
    $\displaystyle R_4[x] = a_0 + a_1x+a_2x^2+a_3x^3 : a_0,...,a_3 \in R$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. linear transformation question
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Mar 19th 2011, 04:10 PM
  2. Linear transformation question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 1st 2010, 02:27 PM
  3. Linear Transformation Question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 4th 2009, 01:51 PM
  4. Linear Transformation Question
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Apr 20th 2008, 09:51 PM
  5. Linear Transformation Question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 6th 2008, 11:23 AM

Search Tags


/mathhelpforum @mathhelpforum