Definitions for linear, affine transformations etc

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:

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


$$ \sum \lambda_i = 1$$

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.


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!!!

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