In linear algebra, I learned that we define linearity as
A map/mapping is linear if it preserved additivity and scaling.
If you prefer symbols:
$L(\vec{\mathbf{v}})$ is linear if:
$$ L(\vec{\mathbf{v}}_1) + L(\vec{\mathbf{v}}_2) = L(\vec{\mathbf{v}}_1 + \vec{\mathbf{v}}_2) $$ $$ L(k\vec{\mathbf{v}}) = kL(\vec{\mathbf{v}})$$
Note: I kind of feel like linearity is one of the fundamental concepts of linear algebra as it’s literally in the name; it’s the algebra of linearity
This implies that mapping a sum of vectors is the same as summing up the mapped individual vectors for the additivity, and
Mapping a scaled vector is the same as scaling the mapped vector.
There are also other things which satisfy additivity and scaling.
A well-known example is the derivative operator $\frac{d}{dx}$, which is linear by the sum rule and constant multiple rule:
$$\frac{d}{dx}(f + g) = \frac{d}{dx}(f) + \frac{d}{dx}(g)$$ $$\frac{d}{dx}(kf) = k\frac{d}{dx}(f)$$
This means that in some sense, mapping a sum of functions is the same as summing up the mapped functions, and that
Mapping a scaled function is the same as scaling the mapped function.
Some other operators that do this are:
- The integral operator
- The Fourier transform operator
- The Laplace transform operator
- The convolution operator
- etc.
The thing that makes linear transformations different is that there is an enlightening geometric interpretation for the linearity of a map:
A linear map $L(\vec{\mathbf{v}})$ is linear if the origin stays fixed and grid lines remain parallel and evenly spaced
If you wanted to get a little formal about the origin part, we can basically say that $L(0) = 0$, where $0$ is the zero vector. This means that a map has to map the origin to the origin to be linear.
Question: Is there any geometric intuition for the linearity of non-linear algebra-ish operators? Is there a thing as “mapping” every function in the space of infinitely differentiate functions to their corresponding derivative?
Any help or idea is appreciated!
This question is related to a question I asked previously: What are some linear operators outside linear algebra or analysis?
A linear map $F: V \to W$ from one linear space $V$ to another linear space $W$ over the same field is nothing more than a statement of the formula $ F L x = L' F x$ , where L and L' are linear endomorphisms of $V,W$ with the same coefficients.
The definition is so general, that its content is nearly emtpy.
Examples start with restriction of a linear map to its kernel, mapped to null space, continues with linear 1-forms on finite dimesional vector spaces for determination of coordinates in a basis, matrix maps between finite dimensional spaces, multilinear maps between tensor products.
Its exploding into an universe of linear functionals over linear subspaces of function spaces with values in abstract linear structures in functional analysis.