easy? proving a linear function

• Mar 31st 2010, 09:02 AM
benjamin872
easy? proving a linear function
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• Mar 31st 2010, 09:11 AM
Defunkt
Quote:

Originally Posted by benjamin872
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 $f(\alpha x) = \alpha x + 3 = 2 * 1 + 3 = 5 \neq \alpha f(x) = \alpha(x+3) = 2(1+3) = 8$
• Mar 31st 2010, 12:55 PM
HallsofIvy
Quote:

Originally Posted by benjamin872
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.
• Mar 31st 2010, 01:04 PM
benjamin872
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
• Apr 1st 2010, 03:17 AM
HallsofIvy
Quote:

Originally Posted by benjamin872
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.