Notation for extension of a linear map Let be $V=V_1\oplus V_2$ a vector space.
Let be $\phi_1$ a linear map on $V_1$.
I would like to define a linear map $\phi_1'$ on $V$ such that for each $v\in V$ we have
$$\tag{1}
\phi_1'(v)=\begin{cases}
\phi_1(v)&\text{if $v\in V_1$},\\
v&\text{if $v\in V_2$},\\
\cdots &\text{otherwise, linearly extended}
\end{cases}
$$
My question is: does the following intuitive notation exist, or is there any other notation about it?
$$\tag{2}
\phi_1'=\phi_1\oplus\operatorname{id}_{V_2}
$$
My goal is to write for a map $\phi$ on $V$ with invariant subspace $V_1$ and $V_2$
$$\tag{3}
\phi=\phi|_{V_1}\oplus\operatorname{id}_{V_2}
+\operatorname{id}_{V_1}\oplus\phi|_{V_2}
$$
EDIT
This would also be something, but not as concise as the other one
$$
\phi=(\iota_{1V}\circ\phi|_{V_1}\circ\pi_1 +
  \iota_{2V}\circ\operatorname{id}_{V_2}\circ\pi_2 )
\circ (\iota_{1V}\circ\operatorname{id}_{V_1}\circ\pi_1 +
  \iota_{2V}\circ\phi|_{V_2}\circ\pi_2 )
$$
 A: This notation is perfectly fine and actually quite common. In fact, it can introduced in a more general context: given vector spaces $V_1,V_2$ and linear maps $f_1\colon V_1\to W$ and $f_2\colon V_2\to W$ one can (uniquely!) define
$$f_1\oplus f_2\colon V_1\oplus V_2\to W,\,(v_1,v_2)\mapsto f_1(v_1)+f_2(v_2)$$
Here, the sum may or not be internal. This is the universal property of the categorical coproduct of vector spaces. The coproduct happens to coincide with the product in the finite case, which is (either by definition of by convention) nothing else than the direct sum. This map does exactly the same as the one you defined by hand.
Also, given two linear maps $f,g\colon V\to W$ there is a natural notion of their sum $f+g$ defined pointwise. This is an interesting obersevation as it gives $\operatorname{Hom}(V,W)$ a very natural (abelian) group structure (with identity the trivial map).
Thus, your map $\phi$ is perfectly valid and there is not need for an inconcise version using composition.
