Fig. 3.2
with equation
x2+y2=1 and the line with equation (see Fig.3.2)
2x=1.
All the coefficients are rational numbers (and even integers). But in the “rational plane”Q2the line does not intersect the circle! Indeed the two “intersection points”
if we can call them that should be 1
2,±
√3 2
—they do not have rational coordinates!
Finally, we seem to end up with the single caseK=R! This is in fact too severe a conclusion, but we shall not insist on the possible generalizations.
Let us conclude this section with a classical notion, valid as soon as the notion of a segment makes sense:
Definition 3.1.2 Let(E, V )be a real affine space. A subsetX⊆Eis convex (see Fig.3.3) when
A, B∈X =⇒ [A, B] ⊆X
i.e.Xcontains the whole segment[A, B]as soon as both extremitiesAandB lie inX.
3.2 Orientation of a Real Affine Space
Let us first introduce the basic ingredient for defining an orientation:
Fig. 3.3
Definition 3.2.1 Let(E, V )be a finite dimensional real affine space. Two affine bases
(O;e1, . . . , en),
O;e1, . . . , en
have the same orientation when in the corresponding change of basis formula
−
→x =M−→x + −→v
the determinant of the matrixMis positive. (See Proposition2.20.1.) The key observation is then:
Proposition 3.2.2 Let(E, V )be a finite dimensional real affine space. The prop- erty “Having the same orientation” is an equivalence relation on the set of affine bases. For non-zero dimensional spaces, there are exactly two equivalence classes.
Proof Under the conditions of Definition3.2.1, the inverse change of basis formula has the form
−
→x =M−1−→ x +−→
v (see Proposition2.20.1). Since
detM−1= 1 detM
we get detM−1>0 and the “having the same orientation” relation is symmetric.
Next if
O;e1, . . . , en ,
O, e1, . . . , en
also have the same orientation, with change of basis formula given by
−
→x=N−→ x + −→w ,
3.2 Orientation of a Real Affine Space 123
Fig. 3.4
Fig. 3.5
then the change of basis formula from the first basis to the third one has the form
−
→x=N M−→x +N−→v + −→w .
But since
detN M=detNãdetM
this quantity remains positive and the “having the same orientation” relation is also transitive, thus it is an equivalence relation.
Replacingenby−en in the first basis yields a change of basis matrix with de- terminant−1: thus these two bases are not in the same equivalence class, proving that there are at least two equivalence classes. Given an arbitrary affine basis, the determinants of the change of basis formulổ to the two bases above (withen and with−en) are opposite real numbers, proving that the arbitrary basis lies in one of
the two equivalence classes.
We can thus make the following definition:
Definition 3.2.3 An orientation of a finite dimensional real affine space consists in choosing one equivalence class of an affine basis, with respect to the equivalence relation “having the same orientation”. The bases in the chosen equivalence class are said to have direct orientation and the bases in the other class, to have inverse orientation.
Keeping in mind the last paragraph of the proof of Proposition3.2.2, let us ex- amine the situation in the 1, 2 and 3-dimensional spaces, with the “usual choice” for the direct orientation.
When one draws the real line horizontally, one generally chooses the left to right orientation as the direct one (Fig.3.4).
In the real plane, turning counter-clockwise, passing frome1toe2without cross- ing the first axis, is generally taken as the direct orientation (Fig.3.5).
Fig. 3.6
In three dimensional space, the direct orientation is generally chosen according to the famous rule of the cork-screw. If you align a cork-screw alonge3and “turn it frome1toe2”, it moves positively in the direction ofe3(see Fig.3.6). A formal translation of this cork-screw rule, in a special case of interest, is given by Exam- ple3.2.4.
Example 3.2.4 Given two linearly independent vectorsx,yinR3, the three vectors x, y, x×y
constitute a basis having the same orientation as the canonical basis.
Proof It suffices to observe that the matrix whose columns are the coordinates ofx, y,x×y has a strictly positive determinant (see Definition3.2.1). Developing this determinant with respect to the third column, we obtain (see Definition1.7.1)
det
x2 y2
x3 y3
2
+det
x1 y1
x3 y3
2
+det
x1 y1
x2 y2
2
.
This quantity is of course positive; it is strictly positive becausexandyare linearly independent, thus at least one of the 2×2 matrices is regular.
Of course in the very special case of the vector spaceRn viewed as an affine space, one generally considers the “canonical basis”
e1=(1,0, . . . ,0), e2=(0,1,0, . . . ,0), . . . , en=(0, . . . ,0,1) as having direct orientation.
The choice of which orientation is to be the direct one is just a matter of personal taste: neither of the two choices is mathematically more “canonical” than the other.
Adopting either orientation will result in an equivalent theory: the only important fact is to keep in mind that both orientations exist simultaneously.
There exists an alternative (and equivalent) topological approach to the notion of orientation (see Problem3.7.4). The setB of all bases of ann-dimensional real vector spaceV can be regarded as a subset ofVnand therefore can be provided with the structure of a metric space. Two bases ofV have the same orientation precisely when they are connected by a continuous path in the metric spaceB.