Image a vector space where two points $P$ and $P^{\prime }$ exist, then there’s a unique translation of the plane that maps $P$ to $P^{\prime }$ which means that the space of translations in the plane can be identified with a set of vectors that exist in the plane, composition of translation correspond to addition of vectors e.g. $\overrightarrow{P{P}^{″}}=\overrightarrow{P{P}^{\prime }}+\overrightarrow{P^{\prime }{P}^{″}}$

An affine space is a space where translation is defined, formally an affine space is a set $E$ (of points) that admits a free transitive action of a vector space $\overrightarrow{E}$ (of translations) whose action results in an element of the set $E$, that is there’s a map $E\times \overrightarrow{E}\to E:(a,\mathbf{v})\mapsto a+\mathbf{v}$ such that

1. The zero vector acts as an identity i.e. for all $a\in E$, $a+\mathbf{0}=a$
2. Addition of vectors correspond to translations i.e. for all $a\in E$ and $\mathbf{u}\mathbf{,}\mathbf{v}\in \overrightarrow{E}$, $x+(\mathbf{u}+\mathbf{v})=(x+\mathbf{u})+\mathbf{v}$
3. For any $a,b\in E$ there’s a unique free vector $\mathbf{u}\in \overrightarrow{E}$ such that $a+\mathbf{u}=b$

The affine space is commonly represented with the triple $⟨E,\overrightarrow{E},+⟩$ where $E$ is a set of points, $\overrightarrow{E}$ a vector space acting on $E$ and an action $+:E\times \overrightarrow{E}\to E$

Consider a subset $L$ of ${\mathbb{A}}^2$ consisting of points satisfying

Where any point has the form $(x,f(x))=(x,2+x)$, the line can be made into an affine space by defining $+:L\times V\to L$ (note that $V$ is a vector space) so that for any $u\in V$

For example the point $(-2,0)$ added with the vector $u=[1,1]$ results in the point $(-1,1)$ which belongs to the set $L$, note that for the example above the vector space $V$ has only vectors parallels to $u=[1,1]$

Chasles’s Identity

Given any three points $a,b,c\in E$ we know that $c=a+\mathbf{ac}$, $b=a+\mathbf{ab}$ and $c=b+\mathbf{bc}$ by the axiom 3, therefore

And thus

Which is known as Chasles’s identity

Affine combinations

Consider ${\mathbb{R}}^2$ an affine space with its origin at $(0,0)$ and basis vectors ${\mathbf{b}}_{\mathbf{1}}=[1,0]$ and ${\mathbf{b}}_{\mathbf{2}}=[0,1]$, given any two points $a,b\in {\mathbb{R}}^2$ with coordinates $a=(a_1,a_2)$ and $b=(b_1,b_2)$ we can define the affine combination $\lambda a+\mu b$ as the point of coordinates

Let $\lambda =1,\mu =1$, $a=(-1,1)$ and $b=(2,2)$ then $a+b=(1,1)$

If we change the coordinate system to have an origin at $(1,1)$ with the same basis vectors then the coordinates of the given points are $a=(-2,-2)$ and $b=(1,1)$, the linear combination is then $a+b=(-1,-1)$ which is the same as the point $(0,0)$ of the first coordinate system, therefore $a+b$ corresponds to two different points depending on the coordinate system used

A restriction is needed for affine combinations to make sense and the restriction is that the scalar add up to 1

Lemma: Given an affine space $E,vE,+$, let $a_i,i\in I$ be a family of points in $E$ and let ${\lambda }_i,i\in I$ a family of scalars then any two points $a,b\in E$ the following properties hold

To prove $\text{(1)}$ we apply Chasles’s identity

For $\text{(2)}$ we also have

Formally for any family of points $a_i,i\in I$ in $E$, for any family ${\lambda }_i,i\in I$ of scalars such that $\sum _{i\in I}{\lambda }_i=1$ the point

Is independent of $a\in E$ and is called the barycenter or affine combination of the points $a_i$ with weights ${\lambda }_i$, and is denoted as

Affine maps

An affine map between two affine spaces $X$ and $Y$ is a map $f:X\to Y$ that preserves affine combinations i.e.