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Math Help - Linear Algebra question I'm stuck on

  1. #1
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    Linear Algebra question I'm stuck on

    Give an example of a function from R^2→R such that

    f(av)=af(v)

    for all a in R and all v in R^2 but f is not linear
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  2. #2
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    Quote Originally Posted by alexmin View Post
    Give an example of a function from R^2→R such that

    f(av)=af(v)

    for all a in R and all v in R^2 but f is not linear
    For \bold{v} = (x,y)\in \mathbb{R}^2 define f(\bold{v})= x. -- This is a mistake.
    Last edited by ThePerfectHacker; January 20th 2008 at 09:33 AM.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    For \bold{v} = (x,y)\in \mathbb{R}^2 define f(\bold{v})= x.
    Sorry, that one doesn't work. The map given by f(\bold{v})= x is linear, and we're looking for a nonlinear map.

    The simplest one I can come up with is the map given by f(x,y) = \left\{\begin{array}{ll}x&\text{if $y\ne0$,}\\ 0&\text{if $y=0$.}\end{array}\right.

    This preserves scalar multiplication: if y≠0 then f(ax,ay)=ax=af(x,y), and if y=0 then f(ax,ay)=f(x,y)=0.
    It is not additive, however, because for example f(1,1) + f(1,–1) = 1+1=2, but (1,1)+(1,–1)=(2,0) and f(2,0) = 0.
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