Every Euclidean space is also a manifold
Euclidean space In mathematics, the term Euclidean space initially describes the space of our perception as it is described in Euclid's elements by axioms and postulates (cf. Euclidean geometry). Up until the 19th century it was assumed that this described the physical space that surrounds us. The addition Euclidean became necessary after more general spatial concepts (e.g. hyperbolic space, Riemannian manifolds) had been developed in mathematics and it became apparent in the context of the special and general theory of relativity that other spatial concepts are required to describe space in physics ( Minkowski space, Lorentz manifold). In the course of time, Euclid's geometry was specified and generalized in different ways: axiomatically by Hilbert (see Hilbert's system of axioms of Euclidean geometry) as Euclidean vector space (a vector space defined by a scalar product) as Euclidean point space (an affine space that models over a Euclidean vector space is) as a coordinate space with the standard scalar product. When Euclidean space is mentioned, each of these can be meant, or a higher-dimensional generalization. The two-dimensional Euclidean space is also called the Euclidean plane. In this two-dimensional case, the term is more generalized in synthetic geometry: Euclidean planes can be defined there as affine planes over a more general class of bodies, the Euclidean bodies. Euclidean space differs from affine space in that lengths and angles can be measured and, accordingly, the images that contain lengths and angles can be distinguished. These are traditionally called congruence maps, other terms are movements and isometries. It differs from hyperbolic space in that the axiom of parallels applies.
Euclidean vector spaces See also: Dot product space
From Euclidean intuition space to Euclidean vector space In analytic geometry one assigns a vector space to Euclidean space. One possibility to do this is to take the amount of parallel displacements (translations), provided with the execution as an addition. Each shift can be described by an arrow that connects a point with its image point. Two arrows, which are parallel in the same direction and are of the same length, describe the same displacement. Two such arrows are called equivalent and the equivalence classes are called vectors. If one selects a point in Euclidean space its position vector as the reference point (origin), then one can assign each point to the point, the vector, which is indicated by an arrow from the origin
is pictured. In this way one gets a one-to-one relationship between the Euclidean space and the associated Euclidean vector space and can thus identify the original Euclidean space with the Euclidean vector space. This identification is not canonical, but depends on the choice of origin. One can now also transfer the length and angle measurements from Euclidean space to vectors as the length of the associated arrows and angles between them. In this way a vector space with a scalar product is obtained. The scalar product is characterized by the fact that the product of a vector with itself is the square of its length. From the mathematical laws for scalar products, the binomial, and formulas and the cosine law (applied to a triangle whose sides the vectors
Euclidean space) the formula results. Here denotes the angle between the vectors and.
General concept Based on this, every real vector space with a scalar product (any finite dimension) is called a Euclidean vector space. The above formula is then used to define the length (norm) of a vector and the angle between vectors. Two vectors are orthogonal if their scalar product is zero. Every three-dimensional Euclidean vector space is isometrically isomorphic to the vector space of the arrow classes. Every -dimensional Euclidean vector space is isometrically isomorphic to the coordinate vector space (see below). Euclidean vector spaces of the same dimension are therefore indistinguishable. This justifies one to denote each such as the Euclidean vector space of the dimension. Some authors also use the term Euclidean space for infinite-dimensional real vector spaces with a scalar product, some also for complex vector spaces with a scalar product, see scalar product space.
Lengths, angles, orthogonality and orthonormal bases See also: Orthonormal bases As soon as one has provided a real vector space with a scalar product, the metric concepts of the Euclidean intuition space can be transferred to it. The length (the norm, the amount) of a vector is then the root of the scalar product of the vector with itself:. Two vectors are orthogonal (or perpendicular) to each other if their scalar product is zero:. The (non-oriented) angle between two vectors is defined using the above formula, i.e. A vector is called a unit vector if it is 1 in length. A basis made up of unit vectors that are pairwise
are orthogonal is called an orthonormal basis. Orthonormal bases exist in every Euclidean vector space. If there is an orthonormal basis, the vector can be represented in this basis:. The coefficients are obtained through.
IsometriesIf and are two-dimensional Euclidean vector spaces, a linear mapping is called a (linear) isometry if it contains the scalar product, i.e. if it applies
. In the case of a self-mapping
an orthogonal map. A
Isometry contains lengths and angles in particular, i.e. especially orthogonality
Conversely, every linear mapping that receives lengths is an isometry. An isometry maps each orthonormal basis back to an orthonormal basis. Conversely, if there is an orthonormal basis of and an orthonormal basis of, then there is exactly one isometry that maps to. It follows from this that two Euclidean vector spaces of the same dimension are isometric, i.e. they cannot be distinguished as Euclidean vector spaces.
The Euclidean Point Space Motivation Euclidean vector spaces often serve as models for Euclidean space. The elements of the vector space are then referred to as points or vectors, depending on the context. No distinction is made between points and their position vectors. This can be computationally advantageous. Conceptually, however, it is unsatisfactory: From a geometric point of view, points and vectors should be conceptually distinguished. Vectors can be added and multiplied by numbers, but points cannot. Points are connected by vectors or merged into one another. There is an excellent element in vector space, the zero vector. In Euclidean geometry, however, all points are equal.
Description The concept of the Euclidean point space provides a remedy. This is an affine space over a Euclidean vector space. A distinction is made here between points and vectors. The totality of the points forms the Euclidean point space. This is usually referred to as. (The superscript is not a Cartesian product.) The totality of all vectors forms a Euclidean vector space, or is not an exponent, but an index that characterizes the dimension. .
For two points each
Exactly one connection vector exists, which is denoted by. The connection vector of a point with itself is the null vector: A point can pass through a vector
can be clearly transferred to a point. This is often referred to as. (This is a purely formal notation. The plus sign does not denote vector space addition, and also no addition on the point space.) The zero vector carries every vector into itself: If the vector leads the point into the point, the point leads into the Point over. This can be done as follows over and the vector in the point
can be expressed:
In the language of algebra these properties mean: The additive group of the vector space transitive on the set.
operates freely and
Lengths, distances and angles The lengths, distances between points, angles and orthogonality can now be defined with the help of the scalar product of vectors: The length of the points and the line and the distance of the
is defined by
The size of the angle
is defined by.
are orthogonal if and only if and are orthogonal.
The angle QPR is the angle between the vectors and
the associated vectors
Representations Length-preserving representations of a Euclidean point space are called isometrics, congruence maps (in plane geometry) or movements. You will also get angles automatically. If there is a movement, then there is an orthogonal mapping (linear isometry), so that for all points and
The real coordinate space definition The -dimensional real coordinate space -Tuple where the is the -fold Cartesian product of the set of real numbers, i.e. the set of real numbers. The elements of the
according to context as points or as vectors, so does not differentiate between points and vectors. As vectors, they are added component by component and multiplied by real numbers: In this case, the elements of the are often written as column vectors (i.e. matrices):
The scalar product (standard scalar product) is defined by. With this scalarp
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