Is $Tf =f(x+4)$ and $Tf=f(x)+4$ a linear map? I have a transformation $T: \mathcal C^0 \rightarrow \mathcal C^0, Tf:=f(x+4)$ and I want to determine if this is linear. As far as I know I need to show that it is closed under addition and scalar multiplication and that it contains the zero element. I am just not sure about the actual "proof".
My attempt:
For it to be linear, $f(u)+f(v)=f(u+v)$ and $f(cu) = cf(u) $ needs to hold. Set $f=u+v$. $T(u)=f(u+1)$. $T(v)=f(v+1)$
$$\implies T(u+v)=f(u+v) \not= f(u)+f(v) \implies \text{not a linear map}$$
Similarly for $f(x)+4$ the map is not linear. Am I applying the critera correctly? In other words. If I am given some transformation or function $f(x)$ do I just "plug in" $(u+v)$ and see if $f(u)+f(v)=f(u+v)$ holds?
 A: Your attempt is, and I mean this in the best possible way, a mess. But that's OK, we live to learn! Here are the problems with your attempt.

*

*You start with "for it to be linear, $f(u)+f(v) = f(u+v)$. This is unclear. What do you mean by "it" in the sentence "for it to be linear"? The condition you write is the condition for $f$ to be linear, but you should be examining the linearity of $T$, not $f$.

*You then say "set $f=u+v$, but what are $u$ and $v$? Are they numbers? Functions? Mappings? Elephants?

*Then you write the equation $T(u) = f(u+1)$ which is completely nonsensical. You do not write where this equation comes from, nor does it make any sense. In order for $T(u)$ to be a sensible expression, $u$ would need to be a function, since $T$ operates on functions, but in order for $f(u+1)$ to be sensible expression, $u$ would need to be a number! So which is it?

*Then you write the even less comprehensible conclusion that $f(u+v)\neq f(u)+f(b)$. Where the heck did this conclusion come from? You never even calculated what $f(u+v)$ is, how can you claim it is not equal to $f(u)+f(v)$? Also, what does the expression $f(u+v)$ even mean? I thought we started with $f=u+v$?

Like I said, a total mess.

In order to properly do the proof, you need to first do a couple of things:

*

*Stand up.

*Take a walk or make yourself some coffee or maybe read a couple of pages of a book.

*Make sure your head is cleared.

OK, done? Now let's get down to business. Follow the steps below:

*

*Write down the exact definition, in words or symbols, of when a mapping is linear. No, that's not just "it's linear when $f(u+v)=f(u)+f(v)$ and $f(cv)=cf(v)$. The exact definition starts with "let $U, V$ be vector spaces and let $M:U\to V$. Then, $M$ is linear if, for all..... and so on.

*Write down, using the above definition, with exact and precise wording, when $T$ in your question is linear.

*Stare at what you wrote down, and if you are still stuck, let me know by writing a comment here (and edit your question with how far you got).

Good luck!
