Are these maps linear maps?

[Frankenstein143]

The task is to decide whether (a),(b),(c),(d),(e) are linear maps or not.
The definition of linear map ist:

L1 F (v + w) = F (v) + F (w),
L2 F (k*v) = k* F (v)

With the definitions I tried to solve (a),(b),(c),(d),(e). Can you check please whether it is correct or not?

Unfortunately, I could not solve (e). Can you help me with that?

 

[pka]

Please define all symbols use in part \(e)\). In particular, \(Abb(\mathbb{R},\mathbb{R})\).

 

[lex]

Unfortunately we don't have the Definition in Aufgabe 3 auf Blatt 5.
Anyway. If you are proving linearity you have to prove both parts of the definition: L1 and L2.
You need to do that for (b) and (d) then.
If you are disproving linearity, all you need is to show that one of L1 and L2 do not hold.
In (a) I think I would like to see more clearly why [MATH](u+v)^3-(u+v)^2≠u^3-u^2+v^3-v^2[/MATH](In other words I would multiply out the brackets and make an argument to show this).
In (c) you have not taken into consideration the 'falls y≠0' etc... conditions, so the argument is not complete.
It is probably easier to disprove L2.
In (e) I assume a polynomial over [MATH]\mathbb{R}[/MATH] is being mapped to a function from [MATH]\mathbb{R}[/MATH] to [MATH]\mathbb{R}[/MATH], with the image being a function mapping x to a value computed by replacing 't' in the polynomial with x.
E.g. [MATH]t^2+t+1 \mapsto \tilde{f}: x \mapsto x^2+x+1 [/MATH]You can then prove that this is linear, by proving both L1 and L2 hold.

 

[Frankenstein143]

There is another Definition that combines L1 and L2 together:
L F (kv + tw) = k* F (v) + t* F (w)
With this Definition I want to prove/disprove the statements:

(e) Yes it is right. For example f=t+1 then f´ x--->x+1. But my question here: Does it matter which function I take? I mean you took t^2+t+1, why you took exactly this function and not another?

(a) I tried to do it again, please look, is it correct now?
(b) and (d) Here I use the Definition L
(c) Here I am really confused because you said that it is easier to disprove L2. But according to my calculation L2 is true?! I also considerate the condition if y≠0. All in all it looks messy to me I do not know whether it is correct or not. If I am wrong again, please give me a solution so that I can compare.
Thank you in advance

 

[lex]

My purpose in taking an example in (e) was not to prove linearity but just to try to explain what I thought the mapping meant.
It is noted that [MATH]\tilde{f}[/MATH] was defined in Example 3 of Sheet 5. Can you show me that definition?
I have written out how I would lay out the solutions.
You are right about (c) - L2 is true! Sorry.

 

[Frankenstein143]

Hey, thank you very much!
here is the Definition of Example 3 in Sheet 5 (I tried to translate it, but I upload the picture too):

Let R be a commutative ring with one. Every polynomial f ∈ R [t] defines a mapping ˜f: R → R through evaluation (“substituting”).
For example, for f = t + 1 ∈ R [t] the mapping ˜f is given by x → x + 1.
Show that the Map A: R [t] → Fig (R, R), f → ˜f with respect to the ring structure defined in the previous exercise f (Abb(R, R), ⊕, *) (so choose X = R in exercise 2) is a homomorphism of rings with one.

I think your first interpretation of (e) was right?

 

[lex]

Thanks yes. That's what I was taking it to be.
Hope my solutions were not too complicated. In (d) and (e) I was trying to write: [MATH]f+\lambda g[/MATH] as a polynomial in [MATH]t[/MATH], i.e. in the form [MATH] \sum_{i=0}^n c_i t^i[/MATH] before doing the mapping. That is why I introduced the rather complicated-looking [MATH]\Sigma[/MATH] notation.