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Homework Help: How can I prove this function is a linear function?

  1. Jul 20, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to prove f(r) = A [itex]\bullet[/itex] (r - kz) is a linear operator, but I don't understand why it is linear.

    2. Relevant equations
    A function of a vector is called linear if:
    f(r1+r2) = f(r1) + f(r2) and f(ar) = af(r)


    3. The attempt at a solution

    f(r1+r2) = A [itex]\bullet[/itex] (r1+r2 - kz) = A[itex]\bullet[/itex]r1 + A[itex]\bullet[/itex]r2 - A[itex]\bullet[/itex]kz

    However, f(r1) + f(r2) = A[itex]\bullet[/itex]r1 - A[itex]\bullet[/itex]kz
    + A[itex]\bullet[/itex]r2 - A[itex]\bullet[/itex]kz

    So, I just proved that the function is NOT a linear function, how great... :(

    By the way, r = (x,y,z) in this book, if that changes anything.
     
  2. jcsd
  3. Jul 20, 2012 #2
    Is it safe to assume this is an error in my book's manual? I even tried developing the vectors, here's my attempt. (Disregard the 3.7.1 part)
     

    Attached Files:

  4. Jul 20, 2012 #3
    Ok, upon further investigation, the only way this can be linear is if I have to apply the -kz part BEFORE writing down my r vectors, which means

    f(r1+r2) = (a1,a2,a3) [itex]\bullet[/itex] (x1+x2,y1+y2,0)

    If that was the case, how in the hell was I supposed to know I had to do that (as opposed to what I did in my scanned solution)??!
     
  5. Jul 20, 2012 #4

    Zondrina

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    Homework Helper

    I'm assuming by you saying r is a vector, you're saying r[itex]\in[/itex]ℝn. Notice that what you're trying to prove is f is closed under addition and scalar multiplication which implies f is linear.

    Suppose that : r1,r2[itex]\in[/itex]ℝn

    Then the way you would go about proving something like f(r1+r2)=f(r1) + f(r2) is to literally ADD the two vectors together since adding two objects together from the same field should result in something from the same field ( If it doesn't then you simply don't have anything to prove here ) and THEN you perform the operator f on the NEW object. Simply separating the newly formed object into its two founding objects is a simple task at this point which should result in your answer.
     
  6. Jul 20, 2012 #5
    k is the unit vector in the z direction, and r is given by:

    r = x i + y j + z k

    Therefore, f (r) = A [itex]\bullet[/itex] (x i + y j)

    Alternately,

    f(r1) = A[itex]\bullet[/itex] (r1 - k z1)

    f(r2) = A[itex]\bullet[/itex] (r2 - k z2)

    Try it now.
     
  7. Jul 21, 2012 #6

    Ray Vickson

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    Homework Helper

    Note: [itex]z[/itex] is the third component of [itex]\textbf{r}[/itex], so if you have [itex]\textbf{r}_1[/itex] and [itex]\textbf{r}_2[/itex] you need [itex]z_1[/itex] and [itex]z_2[/itex].

    RGV
     
  8. Jul 21, 2012 #7
    Yeah thanks everyone. I was assuming the kz wasn't the "same z" as the z component from the r vectors, which means I considered r1-z as (x1,y1,z1)-(0,0,z) = (x1,y1,z1-z).
     
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