Let's say we have a transformation:
$T: \mathbb{R}^n \rightarrow \mathbb{R}^m.$
This is a linear transformation iff: for all $ \vec{a} , \vec{b} \in \mathbb{R}^n$ and $c \in \Bbb{R}$,
- $T(\vec{a} + \vec{b}) = T(\vec{a}) + T(\vec{b})$
- $T(c \vec{a}) = cT(\vec{a})$
I've seen this kind of 'requirements' multiple times in Linear Algebra, and I wonder what the names for these requirements are.