Results 1 to 7 of 7

Math Help - Matrices and Operations

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    9

    Matrices and Operations

    Let A be a n x m matrix. Show that if the function y = f(x) defined for m x 1 matrices x by y = Ax satisfies the linearity property, then f(aw + bz) = af(w) + bf(z) for any real numbers a and b and any m x 1 matrices w and z.



    This is what I did

    Scalars - k1, k2
    Vectors - u, v
    Matrix - n x m matrix

    f(aw + bz) = af(w) + bf(z)
    A(k1u + k2v) = k1Au + k2Av

    m x 1 matrices are the vectors u and v

    multiplying scalars to vectors:
    mk1 x k1 matrix
    mk2 x k2 matrix




    I'm not understanding how I am supposed to prove that this function is linear.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
    Posts
    411
    I don't fully understand your question, or the way the question is given:

    Don't you mean f:\mathbb{R}^m\to \mathbb{R}^n is defined by f(x)= Ax where A is a m\times n matrix?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    9
    Yes, that looks about right, the question itself made me struggle also.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
    Posts
    411
    In that case it's quite trivial:

    f(ax+by) = A(ax+by) =  aAx+ bAy = af(x)+bf(y)

    Matrix miltiplication is linear.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2010
    Posts
    9
    Oh, ok. So say instead of y = Ax it was y = (A^2)x and y = (A^3)x.

    Would the next step in solving them be

    f(ax+by)^2 and f(ax+by)^3 respectively?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466
    No, it would still be linear. The square or cube of a matrix is still a matrix so this is still just "multiplication by a matrix".
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jan 2010
    Posts
    9
    Thanks guys
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Set Operations
    Posted in the Algebra Forum
    Replies: 4
    Last Post: November 6th 2011, 03:55 AM
  2. Replies: 2
    Last Post: November 25th 2010, 06:34 PM
  3. Total matrices and Commutative matrices in GL(r,Zn)
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: August 16th 2010, 02:11 AM
  4. row operations
    Posted in the Algebra Forum
    Replies: 1
    Last Post: July 22nd 2006, 08:26 AM
  5. Replies: 5
    Last Post: February 6th 2006, 03:13 AM

Search Tags


/mathhelpforum @mathhelpforum