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On Wikipedia, the following definition of linear map is given:

Let $V$ and $W$ be vector spaces over the same field $K$. A function $f:V\to W$ is said to be a linear map if for any two vectors $\mathbf{u},\mathbf{v}\in V$ and any scalar $c\in K$, the following conditions are satisfied: \begin{align} f(\mathbf{u}+\mathbf{v})&=f(\mathbf{u})+f(\mathbf{v}) \tag{*}\label{*}\\[5pt] f(c\mathbf{u})&=cf(\mathbf u) \end{align}

My question is: does the $+$ sign on the LHS of $\eqref{*}$ mean something different to the $+$ sign on the RHS? As far as I understand, on the LHS it denotes the vector addition operation of $V$, whereas on the RHS it denotes the vector addition operation of $W$, and these are not necessarily the same operation. Similarly, does $c\mathbf{u}$ refer to scalar multiplication in $V$, whereas $cf(\mathbf{u})$ refers to scalar multiplication in $W$?

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    $\begingroup$ You are absolutely right - in a more precise way one could write $f(u \oplus_V v) = f(u) \oplus_W f(v)$ and $f(c \cdot_V u) = c \cdot_W f(u)$. $\endgroup$
    – Lukas
    Feb 12, 2022 at 20:17
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    $\begingroup$ Yes, if there was a potential for misunderstanding, one might choose to write something like $f(u+_V v)=f(u)+_W f(v)$. Generally, though, context ought to clarify which addition is intended. $\endgroup$
    – lulu
    Feb 12, 2022 at 20:18
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    $\begingroup$ Worth noting: this ambiguity (or, if you prefer, abuse of notation) occurs pretty widely. A group homomorphism, for instance, is required to satisfy $\phi(gh)=\phi(g)\phi(h)$ and nothing in that (standard) notation suggests that the product laws refer to the group laws in two different groups. $\endgroup$
    – lulu
    Feb 12, 2022 at 20:21
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    $\begingroup$ @Asinomás Yes, that's my point. It is just taken for granted that readers will understand that the group law (be it written additively or multiplicatively) is always taken with respect to the relevant domain. $\endgroup$
    – lulu
    Feb 12, 2022 at 20:27
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    $\begingroup$ Your use of the word overloaded perhaps indicates you’re familiar with its usage in computer programming languages. If so, I would say that the meaning of these plus signs is implied by strong type checking. $\endgroup$
    – Deane
    Feb 12, 2022 at 20:54

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You're right. There is a lecturer named Frederic Schuller who has done a mathematical treatment of GR in a set of publicly available Youtube lecture, and in it, he removed the ambiguity by adding subscripts under the addition symbol:

$$ f(u+_V v) = f(u) +_W f(W)$$

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