Definition:Pointwise Addition of Linear Operators

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Definition

Let $V$ be a vector space.

Let $\map \LL V$ denote the set of linear operators on $V$.


$+: \map \LL V \times \map \LL V \to \map \LL V: \forall S, T \in \map \LL V:$
$\forall u \in V: \map {\paren {S + T} } u := \map S u + \map T u$

where $+$ on the right hand side is vector addition.


Specific Instances

Specific instantiations of this concept to particular vector spaces are as follows:


Complex Vector Space

Let $\C^n$ be a complex vector space.

Let $S$ and $T$ be linear operators on $\C^n$.


Then the pointwise sum of $S$ and $T$ is defined as:

$S + T: \C^n \to \C^n:$
$\forall u \in \C^n: \map {\paren {S + T} } u := \map S u + \map T u$

where $+$ on the right hand side is complex vector addition.


Sources