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Math Help - easy? proving a linear function

  1. #1
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    easy? proving a linear function

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    Last edited by benjamin872; April 28th 2010 at 05:26 AM. Reason: .
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  2. #2
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    Quote Originally Posted by benjamin872 View Post
    a linear function has to satisfy:

    additivity f(x+y)=f(x)+f(y)

    and homogeneity f(\alpha x)=\alpha f(x)

    but how do i show this for some simple function like f(x)=x+3??
    i know it must be contained in the domain. but i want to be 100% with my workings im more confused with the whole homogeneous bit.
    It should be clear that this function is not linear; take, say, x=1, y=2 then f(x+y) = x+y+3 = 1+2+3 = 6 \neq f(x) + f(y) = x+3+y+3 = 1+3+2+3 = 9

    and also, if we take, say, \alpha = 2 then f(\alpha x) = \alpha x + 3 = 2 * 1 + 3 = 5 \neq \alpha f(x) = \alpha(x+3) = 2(1+3) = 8
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  3. #3
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    Quote Originally Posted by benjamin872 View Post
    a linear function has to satisfy:

    additivity f(x+y)=f(x)+f(y)

    and homogeneity f(\alpha x)=\alpha f(x)

    but how do i show this for some simple function like f(x)=x+3??
    i know it must be contained in the domain. but i want to be 100% with my workings im more confused with the whole homogeneous bit.
    You don't- as defunkt says, that function is NOT "linear" in this sense. The only linear functions on the real numbers are those of the form f(x)= ax for some a.
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    ok but then il try f(x)=2x+4 say for x=1 and y=2

    i find f(x+y) not equal to f(x)+f(y) which makes 10 not equal 14. so this is not linear also? can someone give me an example of a linear function that works.

    ......... wait so the these properties cannot be satisfied if there is a constant, but i thought a linear function could be plotted y=mx+b im confused
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  5. #5
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    Quote Originally Posted by benjamin872 View Post
    ok but then il try f(x)=2x+4 say for x=1 and y=2

    i find f(x+y) not equal to f(x)+f(y) which makes 10 not equal 14. so this is not linear also? can someone give me an example of a linear function that works.

    ......... wait so the these properties cannot be satisfied if there is a constant, but i thought a linear function could be plotted y=mx+b im confused
    Those are two different meanings of the word "linear". In algebra, a function of the form f(x)= mx+ b is called "linear" because its graph is a straight line.

    But Linear Algebra uses a more stringent definition for "linear transformation"- we must have f(x+ y)= f(x)+ f(y) and f(ax)= af(x). As I said above, the only "linear" functions in that sense, from R to R are of the form f(x)= ax for some number a.
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