Nonlinear Transformations from R to R: Meeting & Breaking Linearity Rules

[randommacuser]

We know that a transformation from V to W is linear if the following hold:
1.) For every x, y in V, T(x+y) = T(x) + T(y)
2.) For every x in V and for every a in R (real numbers), T(ax) = aT(x)

I need two nonlinear transformations from R to R. One must satisfy #1 above and violate #2. The other must violate #1 and satisfy #2.

 

[Hurkyl]

Sounds like homework, so I'm moving it there. What have you done so far on this problem?

 

[dextercioby]

The second one is really simple.Just take a nonlinear operator

$$T(x)=x^{2}$$

I'll let a mathematician deal with the difficult issue.

Daniel.


EDIT:The above is wrong.I'll let a mathematician deal with the whole problem.

 

[Hurkyl]

So are you claiming that T(ax) = aT(x) for your function?

 

[dextercioby]

Ooops,sorry,Hurkyl,didn't see it. That nasty quadratic breaks both if them...

Daniel.

 

[jdavel]

I think T(u) = u + k works for #1 but not #2.

I don't know about the other one.

 

[Data]

No, it doesn't. With that,

$$T(u+v) = u+v+k \neq u+v+2k = T(u) + T(v).$$

 

[Hurkyl]

Here's a hint...

Suppose T(x) satisfies #1, and that you know T(x). Then, you also know T(2x) and T(3x), right? What about T(x/2)? T(47x/163)?

 

[Hurkyl]

PS, this is a standard method of attack, and it's a good way to learn what things "really" mean.

The whole point is to learn precisely what property #1 tells you, so you can find out what you can "break" so that property #2 fails. (and vice versa)

 

[jdavel]

No, it doesn't. With that,

$$T(u+v) = u+v+k \neq u+v+2k = T(u) + T(v).$$

Data,

Touche! What was I thinking?

 

[Hurkyl]

It's a mistake we all make at least once -- the trick is to catch it before you tell anybody.