# Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here.

It requires time to verify statements like below, source here.

According to Wikipedia here, $Ly=f$ and

but I cannot see this kind of statements fast, unless going through the conditions one-by-one. So are there fast ways to check whether some differential is linear or not?

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It's handy to remember that sums of linear operators are linear, so you can break stuff down. –  Alex Becker Sep 20 '12 at 23:24

Before going any further, let's define different types of linearities first. I use this course book here but I provide translations to English below. Feel free to improve or adapt this answer.

Different types of linearity

1. Bilinear form here i.e. $V^k \times W^m\mapsto F^n$ where $F^n$ is a field of scalars -- does $k,m,n$ have some constraint?

2. Sesquilinear, generalizing inner -product to complex numbers here, i.e. $\bar V\times V \mapsto C$.

3. Differential linearity defined on page 319

Work in progress, trying to improve this answer.

1. Do not mix up algebraic linear-independence to linearity of differentials, two different things.
that means "A finite set of functions $\{f_i, i\in\mathbb Z\}\in V$ is algeberaically linear-independent if $\sum_{i=1}^{n}\lambda_if_i=\bf 0 \rightarrow\lambda_i=0,i\in\mathbb Z$."