Linear Operator , visual meaning.

Question

Could someone give me the visual intuition, if there is any, of a Linear Operator $L$ ?

Given the definition that $L$ is linear if:

${L(f+g)=Lf+Lg}$ (with f and g being functions)

and

${L(tf)=tL(f)}$ (with t being scalar)

Answer 1

Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.

Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.

To make another example: Let $V:=\mathbb R^{\mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $S\colon V\to V$, defined by $S\bigl((a_1, a_2,a_3,\dotsc)\bigr)=(0,a_1,a_2,a_3,\dotsc)$.

Answer 2

The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=\frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.