Results 1 to 5 of 5

Math Help - Matrix Transformation problem

  1. #1
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1

    Matrix Transformation problem

    Hi, another question from me again

    A transformation T : R^2 \rightarrow R^2 is represented by the matrix A = \left(\begin{array}{cc}4&4\\4&-2\end{array}\right).

    Find:

    a) The two eigen values for A
    b) A cartesian equation for each of the two lines passing through the origin which are invariant under T.

    a) This was easy enough, -4 and 6.

    b) No idea how to start this, anyone got any pointers?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Invariant means that it does not change under the transformation doesn't it? I'm guessing that you use eigenvalues and eigenvectors at some point?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Sorry just one more thing.

    Could someone please explain what exactly is meant by:

    R^2 \rightarrow R^2
    Not too sure what this means :S

    Thanks again
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    Quote Originally Posted by craig View Post
    Hi, another question from me again

    A transformation T : R^2 \rightarrow R^2 is represented by the matrix A = \left(\begin{array}{cc}4&4\\4&-2\end{array}\right).

    Find:

    a) The two eigen values for A
    b) A cartesian equation for each of the two lines passing through the origin which are invariant under T.

    a) This was easy enough, -4 and 6.

    b) No idea how to start this, anyone got any pointers?

    Thanks
    Quote Originally Posted by craig View Post
    Invariant means that it does not change under the transformation doesn't it? I'm guessing that you use eigenvalues and eigenvectors at some point?
    It means that points on that line are mapped into points on that line. For eigenvectors, v, [tex]Av= \lamba v[/itex] and, interpreting v as position vector that means Av points in the same direction as v: They lie on the same line as v.

    Yes, part b is just asking you to find the eigenvectors corresponding to the eigenvalues and then write the equations of line through the origin in the direction of those vectors. And that's easy: The fact that -4 is an eigenvalue means that there exist non-zero x, y satisfying \begin{bmatrix}4 & 4 \\ 4 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}4x \\ 4y\end{bmatrix}. Multiplying that out and setting components equal immediately gives two equations that both satisfy y= -2x. That is one of the invariant lines.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Thanks for the reply, could you just verify this for me if possible?

    Quote Originally Posted by HallsofIvy View Post
    Yes, part b is just asking you to find the eigenvectors corresponding to the eigenvalues and then write the equations of line through the origin in the direction of those vectors.
    Using -4:

    \left(\begin{array}{cc}4&4\\4&-2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = -4\left(\begin{array}{cc}x\\y\end{array}\right)

    4x + 4y = -4x \rightarrow y = -2x

    And using 6

    \left(\begin{array}{cc}4&4\\4&-2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = 6\left(\begin{array}{cc}x\\y\end{array}\right)

    4x + 4y = 6x \rightarrow y = \frac{1}{2}x

    Would this be correct?

    Thanks again for your time
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Matrix transformation
    Posted in the Algebra Forum
    Replies: 7
    Last Post: July 20th 2010, 07:04 AM
  2. Matrix Transformation
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: July 2nd 2009, 11:17 PM
  3. Matrix of linear transformation, basis problem
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: June 28th 2009, 05:18 PM
  4. Another matrix transformation
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: June 1st 2009, 09:26 AM
  5. Transformation matrix, vector algebra word problem
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 16th 2009, 07:33 PM

Search Tags


/mathhelpforum @mathhelpforum