Linear transformation / Linear map

French: Application Linéaire

A "linear transformation" can be defined by a "linear map" $L$ between two vector spaces $\mathbf V$ and $\mathbf W$ over the same field $\mathbf F$

$$ L : \mathbf V \rightarrow \mathbf W \ L(\vec p) = \vec p' $$

That must preserve addition and scalar multiplication for any vector $\vec u \in \mathbf V$ and $\vec w \in \mathbf V$ and any scalar $c \in \mathbf F$

$$ L(\vec v + \vec w) = L(\vec v ) + L(\vec w) \ L(c \ \vec v) = c \ L(\vec v) $$

Example: a matrix multiplied against a vector is a linear transformation, the matrix itself can be viewed as a linear map $L : \mathbb R^n \rightarrow \mathbb R^n$ between vectors.

Geometric interpretation
A transformation that preserves line parallelism and the origin a the center $L(0) = 0$.

Linear form

A "linear form" is a linear map which image (output) is restricted to a field $\mathbb{F}$:

$L : \mathbf V \rightarrow \mathbb{F} \ $

Example: a map from euclidean vectors to a scalar: $L: \mathbb R^n \rightarrow \mathbb R$

Affine transformation / Affine map

A generalization of an affine transformation is an affine map: given two affine spaces ${\mathcal {A}}$ and ${\mathcal {B}}$, over the same field, a function ${\displaystyle f\colon {\mathcal {A}}\to {\mathcal {B}}}$ is an affine map if and only if for every family ${(x_{i},\lambda_{i})}_{i \in I}$ of weighted points in ${\mathcal {A}}$ such that:

$$ f(\lambda_0 \ x_0 + \lambda_1 \ x_1 + \dots ) = \lambda_0 \ f(x_0) + \lambda_1 \ f(x_1) + \dots \ \ \equiv \ f \left ( \sum \lambda_i \ x_i \right ) = \sum \lambda_i \ f(x_i) \ $$

and

$$ \sum \lambda_i = 1 $$

Geometric interpretation
An affine transformation preserves barycenters.

Affine transformations preserve line parallelism but not lengths and angles. It includes:

• translation
• scaling
• rotation
• shear

Note: Any other combination is an affine transformation:

• Similarity transformation (same shape preserve angles, parallelism, side lengths ratio) are made of uniform scales, rigid transformations and reflections.
• Homothety is when the center of the scale is not at the origin.
• Reflection is performed with change of coordinates and inverse scaling on a single axis.

Linear transformations are a subset of affine transformation where the origin coordinate is always preserved $L(0) = 0$. So, every linear transformation is affine, but not every affine transformation is linear. Affine transformations do not preserve the origin and you can have f((0,0))≠(0,0).

If X is the point set of an affine space, then every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X.

Related

Linear function

In linear algebra a linear map is sometimes called linear function. Which can be confusing since in calculus $f(x) = ax+b$ is also called a linear function although it is a totally different object

Linear function is "linear" because its graph is a line. Linear transformation is linear because it preserves linear structure - the structure of vector space.

Affine combination

An "affine combination" of (x_1, ..., x_n) is a linear combination:

$$\sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \alpha_{1} x_{1} + \alpha_{2} x_{2} + \cdots +\alpha_{n} x_{n}$$

such that

$$\sum_{i=1}^{n} {\alpha_{i}}=1$$

Warning: (\alpha_{i}) not necessarily positive!!!