1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear operator or nonlinear operator?

  1. Jan 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Verify whether or not the operator

    [tex]L(u) = u_x + u_y + 1[/tex]
    is linear.

    2. Relevant equations
    An operator L is linear if for any functions u, v and any constants c, the property

    [tex]L(c_1 u + c_2 v) = c_1 L(u) + c_2 L(v)[/tex]
    holds true.

    3. The attempt at a solution

    I feel as though this should be a linear operator, but the "+1" throws me off as I don't know what linear operator takes any function u, v into 1.

    [tex]L = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + ???[/tex]
    Last edited: Jan 19, 2013
  2. jcsd
  3. Jan 19, 2013 #2


    User Avatar
    Homework Helper

    Alright so, really there's two things you want to verify, but I suppose you could combine them into one condition like that if you want.

    What is [itex]L(c_1 u + c_2 v) = ?[/itex]
  4. Jan 19, 2013 #3
    [tex] L = c_1 \frac{ \partial u}{\partial x} + c_1 \frac{\partial u}{\partial y} + c_2 \frac{ \partial v}{\partial x} + c_2 \frac{\partial v}{\partial y} + ??? [/tex]

    I'm still confused by that constant term.
    Last edited: Jan 19, 2013
  5. Jan 19, 2013 #4


    User Avatar
    Homework Helper

    Just literally write out what the transform would give you, so :

    [itex]L(c_1 u + c_2 v) = c_1u_x + c_1u_y + c_1 + c_2v_x + c_2v_y + c_2[/itex]

    Can you continue from there? Get it into the form [itex]c_1 L(u) + c_2 L(v)[/itex]
  6. Jan 19, 2013 #5
    Sorry for the typos everywhere earlier (edited now).

    [tex]L(c_1 u + c_2 v) = c_1 u_x + c_1 u_y + c_1 + c_2 v_x + c_2 v_y + c_2[/tex]
    [tex]= c_1 (u_x + u_y + 1) + c_2 (v_x + v_y + 1)[/tex]
    [tex]= c_1 L(u) + c_2 L(v)[/tex]

    So is that what you were guiding me to? If I did that correctly, it makes a lot more sense now, thank you. If not...
  7. Jan 19, 2013 #6


    User Avatar
    Science Advisor
    Homework Helper

    No, no, no. [tex]L(c_1 u + c_2 v) = (c_1 u + c_2 v)_x + (c_1 u + c_2 v)_y + 1[/tex]
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Linear operator or nonlinear operator?
  1. Linear operators (Replies: 2)

  2. Linear operator (Replies: 1)

  3. Linear Operator (Replies: 21)