# Linear algebra (Osnabrück 2024-2025)/Part II/Lecture 49

    Orientations on a real vector space
For a one-dimensional real vector space $V$ (s line), every vector $\neq 0$ is a base vector. If we remove the $0$, then the line splits into two halves (two half-lines, two rays). If the line is lying "horizontally“, then we talk about the left-hand and the right-hand side; but this concepts are problematic if we move the line or change the angle of view. However, it is a simple fact that there are two sides, and it is easy to determine whether two points belong to the same side or to different sides, even though it is difficult to put names on the two sides. The two points $\neq 0$ are related by their transformation matrix (base change matrix), which is just one real number. The two points lie on the same side if and only if their transformation matrix is positive, they lie on different sides if and only if their transformation matrix is negative.
In the plane, there is a similar phenomenon, the sense of rotation. If, for example, we want to walk around a tree on the earth, then we can do this clockwise or counter-clockwise. These notions are fixed by the convention that we look "from above“. For a mole in the ground, the sense of rotation is reversed. If we are given a plane in space, it is not at all clear what clockwise means. However, it is for every plane clear that there are two sense of rotation, and whether to such direction coincide. Also this phenomenon can be understood with bases and their transformation matrices. We consider the center of the clock as the origin of the plane, and the finger as a vector, depending on time. For two consecutive points in times $t_{1}$ and $t_{2}$, the finger represents the vectors
$v_{1}$ and $v_{2}$, and we make sure that the finger has moved less than a half rotation. The vectors $v_{1}$ and $v_{2}$ define a basis in the plane (after a half rotation we had $v_{2}=-v_{1}$, and the vectors were linearly dependent). Suppose that at the start time $t_{1}=0$, the finger shows upstairs, then the clockwise motion of the finger is parametrized by ${\begin{pmatrix}\sin t\\\cos t\end{pmatrix}}$.. The two vectors are
    $v_{1}={\begin{pmatrix}0\\1\end{pmatrix}},\,v_{2}={\begin{pmatrix}\sin t\\\cos t\end{pmatrix}}{\text{ with }}0<t<\pi .$
The transformation matrix between the standard basis
    $u_{1}={\begin{pmatrix}1\\0\end{pmatrix}},\,u_{2}={\begin{pmatrix}0\\1\end{pmatrix}}$
is
    $M_{\mathfrak {u}}^{\mathfrak {v}}={\begin{pmatrix}0&\sin t\\1&\cos t\end{pmatrix}}\,.$
The determinant of this matrix is $-\sin t$, and this is negative for $t$ in the given range, for $t$ from $]\pi ,2\pi [$ it is positive. The transformation matrices between the bases
    $v_{1}={\begin{pmatrix}0\\1\end{pmatrix}},\,v_{2}={\begin{pmatrix}\sin t\\\cos t\end{pmatrix}}{\text{ and }}w_{1}={\begin{pmatrix}0\\1\end{pmatrix}},\,w_{2}={\begin{pmatrix}\sin s\\\cos s\end{pmatrix}}{\text{ with }}t,s\in0,\pi$
have a positive determinant. In this way, the clockwise direction does not fix a basis but it determines a class of bases with the property that their transformation matrix always has a positive determinant. For the counter-clockwise direction, we get bases of the type
    $v_{1}={\begin{pmatrix}0\\1\end{pmatrix}},\,v_{2}={\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}{\text{ with }}t\in0,\pi;$
the transformation matrices of these bases to the standard basis have positive determinant.
In space, there is a similar phenomenon; the human body helps to understand it. The right hand and the left hand are mirror-inverted (the leftt hand is the hand that is closer to the heart). We take a hand, we consider the center of the palm as the origin, and we stretch the thumb, the forefinger and the middle finger such that the thumb and the forefinger look like a pistol, and the middle finger points to the inside. The three fingers form in this ordering a sequence of three vectors in space, and they form a basis.
If we do this with the left hand and with the right hand, then it is possible to bring the thumbs and the forefingers in parallel position, but the middle fingers show in opposite direction. The transformation matrix between these two hand bases is
    ${\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}},$
its determinant is negative. The right hand and the left hand represent different orientations of space bases.
We give now the general definition of an orientation. In the following, it is important to recall that a basis is not a set of the basis vectors $\{v_{1},\ldots ,v_{n}\}$ but the ordered tuple $(v_{1},\ldots ,v_{n})$ of basis vectors.

## Definition

Let $V$ denote a real finite-dimensional vector space. Two bases $v_{1},\ldots ,v_{n}$ and $w_{1},\ldots ,w_{n}$ are said to have the same orientation if the determinant of their transformation matrix is
This relation between bases is an equivalence relation; there are only two equivalence classes (called orientation or orientation classes) (with the exception of the zero space).

## Definition

Let $V$ denote a real finite-dimensional vector space. An orientation of $V$ is an equivalence class of bases of $V$ under the equivalence relation of having the
It is easy to determine whether two bases has the same of the opposite orientation; however, it does not make sense to give a name to each orientation.

## Definition

Let $V$ denote a real finite-dimensional vector space. The space is called oriented if an orientation
A vector space becomes oriented by declaring that $V$ has the orientation represented by the basis $v_{1},\ldots ,v_{n}$. The standard space $\mathbb {R} ^{n}$ carries usually the so-called standard orientation, represented by the standard basis $e_{1},\ldots ,e_{n}$. The standard orientation of $\mathbb {R}$ is represented by $1$, and by every positive number. The standard orientation of $\mathbb {R} ^{2}$ corresponds, when we draw the first standard vector to the right, and the second standard vector perpendicular upstairs, to the counter-clockwise orientation. In space, the standard orientation corresponds to the right-hand rule, when the first axis goes to the right, the second axis to the back and the third axis upstairs. The right hand yields a human-natural orientation of the space we are living in, and the standard orientation yields an orientation on $\mathbb {R} ^{3}$; these two things are unrelated, as there are many way to establish a coordinate system. The coordinate system just mentioned rests on a convention.
On an arbitrary real vector space, there is no canonical way to mark an orientation. For every real finite-dimensional vector space a, there exist a bijective linear mapping
    $\varphi \colon \mathbb {R} ^{n}\longrightarrow V,$
and this sends the standard basis to a basis of $V$; however, this image basis and its orientation class depend on the chosen $\varphi$. It is not possible to define on every $V$ an orientation in a canonical way.
On a real affine space $E$, an orientation is just an orientation of the underlying real vector space.

## Definition

Real finite-dimensional vector space/Oriented/Isomorphism/Orientation-preserving/Definition
It is enough to establish this property for just one basis representing the orientation; see exercise *****. For an automorphism $\varphi \colon V\rightarrow V$, we can directly talk about orientation-preserving, without declaring an orientation. We have an orientation-preserving linear mapping if and only if every basis $v_{1},\ldots ,v_{n}$ and its image basis $\varphi (v_{1}),\ldots ,\varphi (v_{n})$ have the same orientation. This follows from the following fact.

## Lemma
### Proof

$\Box$
If $V$ is endowed with an inner product, and $\varphi$ is an isometry, then positive determinant means, due to Lemma 33.13 , that the determinant is $1$. Therefore, a proper isometry is the same as an orientation-preserving isometry.
    Symmetries
The most important evolutionary splitting in the animal kingdom (more precisely, among the eumetazoa) is related to the symmetry of the animals. There are the radial-symmetric animals like jellyfish, and the so-called bilateria. The first ones have only one structuring direction (front and back), along which the animal can develop different features. In the plane perpendicular to this axis, only the distance to the axis makes a difference, they can rotate around the axis. The bilateria have a second structuring direction, beside front and back they have up and down; this makes the development of more complex animals possible. The genetic encoding of the space structure, the basis of this complexification, was an important step in evolution. A consequence was the Cambrian explosion.
Symmetries/Isometries/Cube/Introduction/Section
Symmetries/Isometries/Plane rotations/Introduction/Section
    Footnotes
* ↑ For a $0$-dimensional vector space, that is, the zero space, we only have the empty basis. However, in some situations, it makes sense to talk about two orientations of the zero space, usually denoted by $+$ and $-$.