# Tricky transformations

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.

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

dextercioby
Homework Helper
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.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
So are you claiming that T(ax) = aT(x) for your function?

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

Daniel.

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

I don't know about the other one.

No, it doesn't. With that,

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

Hurkyl
Staff Emeritus
Gold Member
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
Staff Emeritus
Gold Member
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)

Data said:
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
Staff Emeritus