# Question about definition: what is an affine linear space?

In an article I am reading it says : Let $H$ be an affine linear space of codimension $m$...

Could someone please explain me what is meant by affine linear space? Thanks!

• Is $H$ a subset of a bigger space? Because it would be helpful if the answer were 'yes'... – goblin Jun 3 '16 at 13:03
• @goblin I don't think codimension makes any sense otherwise... – Ian Jun 3 '16 at 13:04
• I just stumbled across John Baez's article on torsors, and thought you might find it interesting in connection with this question. As Baez mentions, torsors are to groups as affine spaces are to linear (vector) spaces. – LSpice Sep 7 '16 at 13:47

Others have mentioned how affine spaces arise, which is probably most useful in this context, but it may also be useful to know what they are. (EDIT: As I was writing this definition, @Bernard gave a much punchier version of it; but hopefully the motivation is still useful.)

My favourite example is that points in a fixed Euclidean space form an affine space, and vectors in a fixed Euclidean space form (you guessed it!) a vector space. Note that you can add vectors, but you cannot add points; that there is a canonical 0 vector, but not a canonical 0 point; and that you can subtract points, but the answer is a vector (from the subtrahend to the minuend), not a point. All this is typical of general affine spaces.

Concretely, when one refers to "an affine space $H$", one really means "an affine space $H$ under a vector space $W$". This structure amounts to a translation map $H \times W \to H$, which is often denoted $(h, w) \mapsto h + w$, such that

1. $h + (w_1 + w_2) = (h + w_1) + w_2$ for all $h \in H$ and $w_1, w_2 \in W$,

2. $h + w = h$ if and only if $w = 0$, for all $h \in H$ and $w \in W$, and

3. $h + \cdot$ is a bijection $W \to H$ for all $h \in H$.

The third condition says that, for any two elements $h_1, h_2$ you pick in the affine space, there is a unique vector $w \in W$ such that $h_1 + w = h_2$. (The first says that the two compositions $(H \times W) \times W \to H \times W \to H$ and $H \times (W \times W) \to H \times W \to H$ agree, i.e., that a certain diagram commutes; or, more briefly, that the translation map is a set action on $H$ of the additive group underlying $W$.) If one writes (suggestively) $w = h_2 - h_1$ for the unique solution to this equation, then one can equivalently think of an affine space as being equipped with the map $H \times H \to W$ given by $(h_1, h_2) \mapsto h_2 - h_1$; this is the same data, and then our conditions become

1. $(h_3 - h_2) + (h_2 - h_1) = h_3 - h_1$ for all $h_1, h_2, h_3 \in H$,

2. $h_2 - h_1 = 0$ if and only if $h_1 = h_2$ for all $h_1, h_2 \in H$, and

3. $\cdot - h$ is a bijection $H \to W$ for all $h \in H$.

To make $\{\text{points of a fixed Euclidean space}\}$ into an affine space under $\{\text{vectors of that same Euclidean space}\}$, define $P + \vec v$ to be $Q$, where $Q$ is the endpoint of the vector $\vec v$ when it is translated so that its start is at $Q$. Again, it may be easier to think of the subtraction map $P - Q = \overrightarrow{QP}$, which subtracts points and produces vectors. (Another example in the same spirit is that dates form an affine space under times: you can add times, but not dates; there is a canonical 0 time but not a canonical 0 date; and you can subtract dates to get the amount of time between them.)

Note that you can always make an affine space $H$ under $W$ into a vector space (in fact a bijective copy of $W$); just pick any point $h_0 \in H$ to treat as the origin, and then define, for $h_1, h_2 \in H$ and $\lambda$ in the underlying field, $h_1 + h_2$ to be $h_0 + (w_1 + w_2)$ and $\lambda h_1$ to be $h_0 + \lambda w_1$, where $w_i = h_i - h_0$. It is in this sense that many people will say that an affine space is a vector space in which you have "forgotten the origin"; one might say as well say that it is a vector space in which any point can be chosen to be the origin.

If $V$ is a vector space, $W \subseteq V$ is a subspace, and $H \subseteq V$ is an affine space under $W$ where the structure map $H \times W \to H$ is the usual vector addition, then it is easy to show that, for any vector $h \in H$, we have that $H = h + W$. That is, $H$ is a coset of a subspace of $V$. Indeed, as @PedroTamaroff mentions, this is sometimes taken as the definition of an affine subspace of a vector space. In this setting, the dimension and codimension of $H$ are, by fiat, the same as those of $W$.

• A better answer than mine, and IMO also a better answer than a fully abstract one. +1. – Ian Jun 3 '16 at 13:21
• @Ian, on the other hand, your discussion of the way that these arise (as solutions to inhomogeneous linear equations) should not be missed as motivation for the concept. – LSpice Jun 3 '16 at 13:23
• @PedroTamaroff It is a bit odd from the perspective of modern linear algebra because we don't have a standard notation for "$\mathbb{R}^n$ the affine space". Call it $\mathbb{A}^n$ for ease of discussion. Then for whatever choice of origin point you like, you have a mapping from $\mathbb{A}^n$ to $\mathbb{R}^n$ given by "draw the vector from the origin you chose to the point". Regardless of which origin you chose, you can define subtraction by mapping two points to their corresponding vectors and then subtracting the vectors, making sure that you kept the origin the same for both points. – Ian Jun 3 '16 at 13:32
• @ElijahRockers , I think that your nice application may be formalised mathematically with the observation that choosing an origin in $\mathbb A^3$ gives an identification of the affine endomorphism of $\mathbb A^3$ with $\mathrm{GL}_3 \ltimes \mathbb A^3$. Your observation that there are 12 degrees of freedom corresponds to the fact that $\mathrm{GL}_3 = \mathrm{SO}_3\cdot\mathrm{GL}_1^3\cdot U$, where $U$ is the upper uni-triangular matrices, is 9-dimensional, and then, of course, we pick up 3 additional dimensions of translations. – LSpice Jun 3 '16 at 17:55
• @Ian, you can write $\mathbb{R}^3_\mathbf{CRing}, \mathbb{R}^3_{\mathbb{R}\mathbf{Mod}}, \mathbb{R}^3_{\mathbb{R}\mathbf{Aff}}$ to distinguish these. – goblin Jun 4 '16 at 4:44

An affine space looks geometrically like a vector space except that you might have moved the origin to another point. The classic example is the solution set to a linear equation $Ax=b$; when $b=0$ this is a vector space, when $b \neq 0$ it is not a vector space but it is an affine space. The notion of affine space also buys us some algebra, albeit different algebra from the usual vector space algebra. The algebra is different because there is no natural way to add vectors in an affine space, but there is a natural way to subtract them, producing vectors called displacement vectors that live in the vector space associated to our affine space.

The abstract definition is that, given a vector space $E$, an affine space $\mathcal E$ with direction $E$ is a non-empty set $\mathcal E$ with a free and transitive action of the additive group of $E$ on $\mathcal E$.

"Let H be an affine linear space of codimension $m$"

what is meant by affine linear space?

There are two related meanings: $H$ is not required to contain $0$, and $H$ is not required to have any of its points distinguished as an "origin". For the phrase in the question to make sense, $H$ should be inside a linear space that has a distinguished origin.

In practice it means $H$ is cut out by linear equations that are allowed to have nonzero constant terms, or is the image of a non-homogeneous vector-valued linear function $Ax + b$. The minimum number of inhomogeneous linear equations necessary to define $H$ is the codimension $m$.

• This is probably the most useful way of thinking about the codimension. Although it is probably clearly implicit, it may be useful to mention that an affine subspace isn't just any random subset not necessarily containing $0$; it must rather satisfy some conditions (amounting to requiring that it be a coset of a vector subspace). – LSpice Jun 3 '16 at 20:13

Definition: Let $V$ a vectorial space. A set $B$ (also $\emptyset$), is called affine space if $\exists F: B\times B \to V$ such that $\forall P, Q \in A, F(P,Q)= v \in V$ with following properties:

1. $\forall P \in B, \forall v \in V, \exists! Q \in B$ such that $v=F(P,Q).$
2. $\forall P,Q,R \in B, F(P,Q) +F(Q,R)=F(P,R).$ (Chasles's relation).

We can define the dimension of an affine space in this way: $dim(B)=dim(V)$ if $A \neq \emptyset$ and $dim(B)=-1$ if $B=\emptyset$.