Results 1 to 4 of 4

Math Help - Linear Transformations (again)

  1. #1
    Newbie
    Joined
    May 2010
    From
    chapel hill, nc
    Posts
    4

    Linear Transformations (again)

    V is the vector space of functions that have continuous 2nd derivatives on (2, 5). (meaning y' and y are both also continuous on (2, 5)). W is the vector space of continuous functions on (2, 5).

    a) L is the function that assigns to each such "input" function y the "output" function y'y. For each input function y, the output function is called L(y). Then L(y)=y'y, which means that for every t allrealnumbers (2, 5), L[y](t) = y'(t)y(t). For example, if y(t)=t^3 then L[y](t) = (3t^3)(t^3)

    b) L is the function that assigns to each such "input" function y the "output" function 4y'' - 3y' + 2y. For each input function y, the output function is called L[y]. Then L[y] = 4y'' - 3y' + 2y, which means that for every t allrealnumbers (2, 5), L[y](t) = 4y''(t) -3y'(t) + 2y(t)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,543
    Thanks
    1394
    Quote Originally Posted by tanelly View Post
    V is the vector space of functions that have continuous 2nd derivatives on (2, 5). (meaning y' and y are both also continuous on (2, 5)). W is the vector space of continuous functions on (2, 5).

    a) L is the function that assigns to each such "input" function y the "output" function y'y. For each input function y, the output function is called L(y). Then L(y)=y'y, which means that for every t allrealnumbers (2, 5), L[y](t) = y'(t)y(t). For example, if y(t)=t^3 then L[y](t) = (3t^3)(t^3)

    b) L is the function that assigns to each such "input" function y the "output" function 4y'' - 3y' + 2y. For each input function y, the output function is called L[y]. Then L[y] = 4y'' - 3y' + 2y, which means that for every t allrealnumbers (2, 5), L[y](t) = 4y''(t) -3y'(t) + 2y(t)
    ??? I don't see a question here! Are you asking which of these is a linear transformation? Is L(x^2+ x^3)= L(x^2)+ L(x^3) in each case?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    From
    chapel hill, nc
    Posts
    4
    I apologize. The question is the same as the other post I made. By checking both i and ii (as follows), determine whether of not the function L is really a linear transformation.

    vectors denoted by bold

    i) for every m and n in V, L(m+n)=L(m) + L(n)
    ii) for every m in V and every scalar c, L(cm) = c(L(m))
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,543
    Thanks
    1394
    Okay, for the first one, the question can be answered by looking, as I suggested, at L(x^3+ x^2) and L(x^3)+ L(x^2). Are they the same?

    For the second one, you have to be more general. Look at L(af(x)+ bg(x) where a and b are numbers, f and g are twice differentiable functions. L(af(x)+ bg(x))= 4(af(x)+ bg(x))"- 3(af(x)+ bg(x))'+ 2(af(x)+ bg(x)). Is that equal to aL(f(x))+ bL(g(x))= a(4f''(x)- 3f'(x)+ 2f(x))+ b(4g''(x)- 3g'(x)+ 2g(x))?

    (My saying that you can use a single example for the first but that the second requires you to be "more general" sort of gives the answer away!)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Transformations and the General Linear Group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 26th 2011, 10:50 AM
  2. Linear Map transformations (Linear Algebra)
    Posted in the Algebra Forum
    Replies: 4
    Last Post: October 21st 2011, 09:56 AM
  3. Basic Linear Algebra - Linear Transformations Help
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 7th 2010, 03:59 PM
  4. Linear Independence in linear transformations
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 17th 2009, 04:22 PM
  5. Replies: 3
    Last Post: June 2nd 2007, 10:08 AM

Search Tags


/mathhelpforum @mathhelpforum