Are these maps linear maps?

Frankenstein143

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May 17, 2021
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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?



LINABBILDUNG.jpg
 

pka

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Jan 29, 2005
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Please define all symbols use in part \(e)\). In particular, \(Abb(\mathbb{R},\mathbb{R})\).
 

lex

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Mar 3, 2021
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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 \(\displaystyle (u+v)^3-(u+v)^2≠u^3-u^2+v^3-v^2\)
(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 \(\displaystyle \mathbb{R}\) is being mapped to a function from \(\displaystyle \mathbb{R}\) to \(\displaystyle \mathbb{R}\), with the image being a function mapping x to a value computed by replacing 't' in the polynomial with x.
E.g. \(\displaystyle t^2+t+1 \mapsto \tilde{f}: x \mapsto x^2+x+1 \)
You can then prove that this is linear, by proving both L1 and L2 hold.
 
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