Results 1 to 4 of 4

Math Help - Linear transformation of the line

  1. #1
    Newbie
    Joined
    Oct 2011
    Posts
    1

    Linear transformation of the line

    So I have this question:

    Find linear transformations of R2 which satisfy the following conditions: Sends the line y = 4x to the origina, but isn't the zero map.

    I know that my teacher doesn't want us to use [0, 0, 0, 0] but rather find a transformation that sends the line back to the origin of the plane.

    so y = 4x can become y - 4x = 0, so I would have to make both y and x 0 to make them collapse to the origin, right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,286
    Thanks
    673

    Re: Linear transformation of the line

    no. you would have to send all points (x,4x) to (0,0). it stands to reason that the expression y-4x should figure somewhere in the definition of your transformation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298

    Re: Linear transformation of the line

    Any linear transformation from R^2 to R^2 can be written as
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}
    The fact that the entire line y= 4x is mapped to (0, 0) means that
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}x \\ 4x\end{bmatrix}= \begin{bmatrix}ax+ 4bx \\ cx+ 4dx\end{bmatrix}= \begin{bmatrix}0 \\ 0\end{bmatrix}

    which gives the two equations ax+ 4bx= 0 and cx+4dx= 0, for all x. Dividing through by x, a+ 4b= 0, c+ 4d= 0. Obviously a= b= c= d= 0 but that is not an acceptable soution. Instead write the equations as a= -4b, c= -4d. Choose any non-zero values you want for b and d.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,286
    Thanks
    673

    Re: Linear transformation of the line

    i vote for b = d = 1, because 1 is my favoritest number in the hole-wide world.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 1st 2011, 10:00 PM
  2. Replies: 1
    Last Post: February 27th 2011, 03:02 AM
  3. Replies: 1
    Last Post: June 9th 2009, 10:22 AM
  4. Replies: 2
    Last Post: May 2nd 2009, 05:18 PM
  5. Linear Algebra.Linear Transformation.Help
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 5th 2009, 01:14 PM

Search Tags


/mathhelpforum @mathhelpforum