Results 1 to 6 of 6

Thread: Linear Transformation

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    19

    Linear Transformation

    Prove whether or not these functions are linear transformation.

    a) $\displaystyle F:V_3(R) \to V_2(R),$ defined by $\displaystyle F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)$

    b) $\displaystyle G:V_3(R) \to V_2(R),$ defined by $\displaystyle G(a_1, a_2, a_3)=(a_1-a_2, 2a_1-a_3)$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,724
    Thanks
    3008
    A transformation, f, is linear if and only if it satisfies f(au+ bv)= af(u)+ bf(v) for any vectors u and v and any numbers a and b.

    If $\displaystyle F(a_1, a_2, a_3)= (a_1+ a_2, a_1a_3- a_2)$ then what is $\displaystyle F(a_1+ b_1, a_2+ b_2, a_3+ b_3)$. Is it the same as $\displaystyle F(a_1,a_2,a_3)+ F(b_1, b_2, b_3)$?
    Last edited by mr fantastic; May 4th 2009 at 03:17 AM. Reason: Fixed a subscript
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    19
    Yeah I don't know how to show whether it is Linear or not.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    9
    Quote Originally Posted by LL_5 View Post
    Yeah I don't know how to show whether it is Linear or not.
    You have been told what to do. Try answering the question you were asked in post #2. Where do you get stuck?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2008
    Posts
    19
    This is what I have done so far to prove whther it is linear:
    Attached Thumbnails Attached Thumbnails Linear Transformation-new.bmp  
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    $\displaystyle

    F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)
    $

    Let $\displaystyle v_1= (x_1,y_1,z_1)$ and $\displaystyle v_2=(x_2,y_2,z_2)$

    A map is linear if $\displaystyle F(\alpha_1 v_1+\alpha_2 v_2)=\alpha_1F(v_1)+\alpha_2F(v_2)$

    $\displaystyle F \left( \alpha_1 (x_1,y_1,z_1)+\alpha_2(x_2,y_2,z_2) \right)=F(\alpha_1 x_1+ \alpha_2 x_2, \alpha_1y_1+\alpha_2y_2, \alpha_1 z_1+ \alpha_2 z_2)$$\displaystyle =(\alpha_1 x_1+\alpha_2 x_2 +\alpha_1 y_1+\alpha_2 y_2, \alpha_1^2 x_1 z_1 + \alpha_2 \alpha_1 x_2 z_1+ \alpha_1 \alpha_2 x_1 z_2 + \alpha_2^2 x_2 z_2-\alpha_1y_1-\alpha_2y_2)$

    $\displaystyle \alpha_1F(v_1)+\alpha_2F(v_2)=\alpha_1(x_1+y_1, x_1 z_1-y_1)+ \alpha_2 (x_2+y_2, x_2 z_2-y_2)$

    But $\displaystyle F(\alpha_1 v_1+\alpha_2 v_2) \neq \alpha_1F(v_1)+\alpha_2F(v_2)$ so $\displaystyle F$ is not a linear transformation.

    Now you try and do $\displaystyle G$. I think it is linear.....
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Aug 1st 2011, 10:00 PM
  2. Example of a linear transformation T:R^4 --> R^3?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 5th 2011, 07:04 PM
  3. linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 28th 2009, 06:40 AM
  4. Linear Algebra.Linear Transformation.Help
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 5th 2009, 01:14 PM
  5. Linear Transformation
    Posted in the Pre-Calculus Forum
    Replies: 10
    Last Post: May 25th 2008, 12:14 AM

Search Tags


/mathhelpforum @mathhelpforum