🆕 Wall Street Veteran says to “back up the truck” on this stock ⬇ | February 28

But Wall Street veteran Whitney Tilson says you don't need to own dozens and dozens of stocks to do well in the markets. In fact, he believes you could make massive gains - and worry far less about your finances - if you focus on finding one truly great stock idea and "back up the truck." [RelaxAndTrade]( Occasionally, an opportunity comes to our attention at Relax And Trade we believe readers like you will find valuable. The message below from one of our partners is one we believe you should take a close look at. Dear Reader, The average investor often owns far too many stocks. But Wall Street veteran Whitney Tilson says you don't need to own dozens and dozens of stocks to do well in the markets. In fact, he believes you could make massive gains - and worry far less about your finances - if you focus on finding one truly great stock idea and "back up the truck." In fact, he has one such high-conviction idea right now... He's so sure of it, he says he'd put half of his kid's college fund into it "without blinking an eye." He calls it "[America's #1 Retirement Stock](. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension,[1] including the three-dimensional space and the Euclidean plane (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements,[2] with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate). After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article.[3] In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space � � , {\displaystyle \mathbb {R} ^{n},} equipped with the dot product. An isomorphism from a Euclidean space to � � \mathbb {R} ^{n} associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point. Definition History of the definition Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry. In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension.[4] Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces. Motivation of the modern definition One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below). In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres. The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts, the space of translations which is equipped with an inner product.[1] The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles. The set � � \mathbb {R} ^{n} of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and � � \mathbb {R} ^{n} viewed as a Euclidean space. It follows that everything that can be said about a Euclidean space can also be said about � � . \mathbb{R} ^{n}. Therefore, many authors, especially at elementary level, call � � \mathbb {R} ^{n} the standard Euclidean space of dimension n,[5] or simply the Euclidean space of dimension n. A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of � � \mathbb {R} ^{n} is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no origin nor any basis in the physical world. Technical definition A Euclidean vector space is a finite-dimensional inner product space over the real numbers.[6] A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces.[6] If E is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted � → . {\displaystyle {\overrightarrow {E}}.} The dimension of a Euclidean space is the dimension of its associated vector space. The elements of E are called points and are commonly denoted by capital letters. The elements of � →{\displaystyle {\overrightarrow {E}}} are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting of the action of a Euclidean vector on the Euclidean space. The action of a translation v on a point P provides a point that is denoted P + v. This action satisfies � + ( � + � ) = ( � + � ) + � . {\displaystyle P+(v+w)=(P+v)+w.} Note: The second + in the left-hand side is a vector addition; all other + denote an action of a vector on a point. This notation is not ambiguous, as, for distinguishing between the two meanings of +, it suffices to look on the nature of its left argument. The fact that the action is free and transitive means that for every pair of points (P, Q) there is exactly one displacement vector v such that P + v = Q. This vector v is denoted Q − P or � � → . {\displaystyle {\overrightarrow {PQ}}.} As previously explained, some of the basic properties of Euclidean spaces result of the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure and its subsections. Prototypical examples For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space. A typical case of Euclidean vector space is � � \mathbb {R} ^{n} viewed as a vector space equipped with the dot product as an inner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space E of dimension n, the choice of a point, called an origin and an orthonormal basis of � →{\displaystyle {\overrightarrow {E}}} defines an isomorphism of Euclidean spaces from E to � � . \mathbb{R} ^{n}. As every Euclidean space of dimension n is isomorphic to it, the Euclidean space � � \mathbb {R} ^{n} is sometimes called the standard Euclidean space of dimension n.[5] Affine structure Main article: Affine space Some basic properties of Euclidean spaces depend only of the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections. Subspaces Main article: Flat (geometry) Let E be a Euclidean space and � →{\displaystyle {\overrightarrow {E}}} its associated vector space. A flat, Euclidean subspace or affine subspace of E is a subset F of E such that � → = { � � → ∣ � ∈ � , � ∈ � } {\displaystyle {\overrightarrow {F}}=\left{{\overrightarrow {PQ}}\mid P\in F,Q\in F\right\}} as the associated vector space of F is a linear subspace (vector subspace) of � → . {\displaystyle {\overrightarrow {E}}.} A Euclidean subspace F is a Euclidean space with � →\overrightarrow F as the associated vector space. This linear subspace � →\overrightarrow F is also called the direction of F. If P is a point of F then � = { � + � ∣ � ∈ � → } . {\displaystyle F=\left\{P+v\mid v\in {\overrightarrow {F}}\right\}.} Conversely, if P is a point of E and � →{\displaystyle {\overrightarrow {V}}} is a linear subspace of � → , {\displaystyle {\overrightarrow {E}},} then � + � = { � + � ∣ � ∈ � } {\displaystyle P+V=\left\{P+v\mid v\in V\right\}} is a Euclidean subspace of direction � →{\displaystyle {\overrightarrow {V}}}. (The associated vector space of this subspace is � →{\displaystyle {\overrightarrow {V}}}.) A Euclidean vector space � →{\displaystyle {\overrightarrow {E}}} (that is, a Euclidean space that is equal to � →{\displaystyle {\overrightarrow {E}}}) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector. Lines and segments In a Euclidean space, a line is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form { � + � � � → ∣ � ∈ � } , {\displaystyle \left\{P+\lambda {\overrightarrow {PQ}}\mid \lambda \in \mathbb {R} \right\},} where P and Q are two distinct points of the Euclidean space as a part of the line. It follows that there is exactly one line that passes through (contains) two distinct points. This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through P and Q is { � + ( 1 − � ) � � → + � � � → ∣ � ∈ � } , {\displaystyle \left\{O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mid \lambda \in \mathbb {R} \right\},} where O is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for O; this allows simplifying the preceding formula into { ( 1 − � ) � + � � ∣ � ∈ � } . {\displaystyle \left\{(1-\lambda )P+\lambda Q\mid \lambda \in \mathbb {R} \right\}.} A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter. The line segment, or simply segment, joining the points P and Q is the subset of points such that 0 ≤ 𝜆 ≤ 1 in the preceding formulas. It is denoted PQ or QP; that is � � = � � = { � + � � � → ∣ 0 ≤ � ≤ 1 } . {\displaystyle PQ=QP=\left\{P+\lambda {\overrightarrow {PQ}}\mid 0\leq \lambda \leq 1\right\}.} Parallelism Two subspaces S and T of the same dimension in a Euclidean space are parallel if they have the same direction (i.e., the same associated vector space).[a] Equivalently, they are parallel, if there is a translation vector v that maps one to the other: � = � + � . {\displaystyle T=S+v.} Given a point P and a subspace S, there exists exactly one subspace that contains P and is parallel to S, which is � + � → . {\displaystyle P+{\overrightarrow {S}}.} In the case where S is a line (subspace of dimension one), this property is Playfair's axiom. It follows that in a Euclidean plane, two lines either meet in one point or are parallel. The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other. Metric structure The vector space � →{\displaystyle {\overrightarrow {E}}} associated to a Euclidean space E is an inner product space. This implies a symmetric bilinear form � → × � → → � ( � , � ) ↦ ⟨ � , � ⟩{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}} that is positive definite (that is ⟨ � , � ⟩\langle x,x\rangle is always positive for x ≠0). The inner product of a Euclidean space is often called dot product and denoted x ⋅ y. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is ⟨ � , � ⟩\langle x,y\rangle will be denoted x ⋅ y in the remainder of this article. The Euclidean norm of a vector x is ‖ � ‖ = � ⋅ � . {\displaystyle \|x\|={\sqrt {x\cdot x}}.} The inner product and the norm allows expressing and proving metric and topological properties of Euclidean geometry. The next subsection describe the most fundamental ones. In these subsections, E denotes an arbitrary Euclidean space, and � →{\displaystyle {\overrightarrow {E}}} denotes its vector space of translations. Distance and length Main article: Euclidean distance The distance (more precisely the Euclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is � ( � , � ) = ‖ � � → ‖ . {\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\vphantom {\frac {|}{}}}{\Bigr \|}.} The length of a segment PQ is the distance d(P, Q) between its endpoints P and Q. It is often denoted | � � | {\displaystyle |PQ|}. The distance is a metric, as it is positive definite, symmetric, and satisfies the triangle inequality � ( � , � ) ≤ � ( � , � ) + � ( � , � ) . {\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).} Moreover, the equality is true if and only if a point R belongs to the segment PQ. This inequality means that the length of any edge of a triangle is smaller than the sum of the lengths of the other edges. This is the origin of the term triangle inequality. With the Euclidean distance, every Euclidean space is a complete metric space. Orthogonality Main articles: Perpendicular and Orthogonality Two nonzero vectors u and v of � →{\displaystyle {\overrightarrow {E}}} (the associated vector space of a Euclidean space E) are perpendicular or orthogonal if their inner product is zero: � ⋅ � = 0 {\displaystyle u\cdot v=0} Two linear subspaces of � →{\displaystyle {\overrightarrow {E}}} are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector. Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said perpendicular. Two segments AB and AC that share a common endpoint A are perpendicular or form a right angle if the vectors � � →{\displaystyle {\overrightarrow {AB}}} and � � →{\displaystyle {\overrightarrow {AC}}} are orthogonal. If AB and AC form a right angle, one has | � � | 2 = | � � | 2 + | � � | 2 . {\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.} This is the Pythagorean theorem. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product: | � � | 2 = � � → ⋅ � � → = ( � � → + � � → ) ⋅ ( � � → + � � → ) = � � → ⋅ � � → + � � → ⋅ � � → − 2 � � → ⋅ � � → = � � → ⋅ � � → + � � → ⋅ � � → = | � � | 2 + | � � | 2 . {\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}\\&=\left({\overrightarrow {BA}}+{\overrightarrow {AC}}\right)\cdot \left({\overrightarrow {BA}}+{\overrightarrow {AC}}\right)\\&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\&=|AB|^{2}+|AC|^{2}.\end{aligned}}} Here, � � → ⋅ � � → = 0 {\displaystyle {\overrightarrow {AB}}\cdot {\overrightarrow {AC}}=0} is used since these two vectors are orthogonal. Angle Main article: Angle Positive and negative angles on the oriented plane The (non-oriented) angle θ between two nonzero vectors x and y in � →{\displaystyle {\overrightarrow {E}}} is � = arccos ⁡ ( � ⋅ � ‖ � ‖ ‖ � ‖ ) {\displaystyle \theta =\arccos \left({\frac {x\cdot y}{\left\|x\right\|\left\|y\right\|}}\right)} where arccos is the principal value of the arccosine function. By Cauchy–Schwarz inequality, the argument of the arccosine is in the interval [−1, 1]. Therefore θ is real, and 0 ≤ θ ≤ π (or 0 ≤ θ ≤ 180 if angles are measured in degrees). Angles are not useful in a Euclidean line, as they can be only 0 or π. In an oriented Euclidean plane, one can define the oriented angle of two vectors. The oriented angle of two vectors x and y is then the opposite of the oriented angle of y and x. In this case, the angle of two vectors can have any value modulo an integer multiple of 2π. In particular, a reflex angle π Over his 20-plus years running a hedge fund firm, Whitney made the biggest gains for his investors by nailing the timing on these "back up the truck" opportunities... This strategy allowed him to triple his investors' money in a flat market in his fund's first decade, growing his hedge fund from $1 million to $200 million. He's been nicknamed "The Prophet" by CNBC for his accurate market predictions spanning the last two decades. Today, Whitney sees an incredible "back up the truck" opportunity in “America's #1 Retirement Stock.” He recently put together a free presentation where you can learn the name and ticker symbol of this company. He's doing so because he knows millions of Americans out there are completely unprepared for retirement and need to catch up quickly - but also with less risk. You can watch his full presentation and discover his favorite stock idea, completely free, by [clicking here](. [Press play]( Cartesian coordinates See also: Cartesian coordinate system Every Euclidean vector space has an orthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a basis ( � 1 , … , � � ) {\displaystyle (e_{1},\dots ,e_{n})} of unit vectors ( ‖ � � ‖ = 1 {\displaystyle \|e_{i}\|=1}) that are pairwise orthogonal ( � � ⋅ � � = 0 {\displaystyle e_{i}\cdot e_{j}=0} for i ≠j). More precisely, given any basis ( � 1 , … , � � ) , {\displaystyle (b_{1},\dots ,b_{n}),} the Gram–Schmidt process computes an orthonormal basis such that, for every i, the linear spans of ( � 1 , … , � � ) {\displaystyle (e_{1},\dots ,e_{i})} and ( � 1 , … , � � ) {\displaystyle (b_{1},\dots ,b_{i})} are equal.[7] Given a Euclidean space E, a Cartesian frame is a set of data consisting of an orthonormal basis of � → , {\displaystyle {\overrightarrow {E}},} and a point of E, called the origin and often denoted O. A Cartesian frame ( � , � 1 , … , � � ) {\displaystyle (O,e_{1},\dots ,e_{n})} allows defining Cartesian coordinates for both E and � →{\displaystyle {\overrightarrow {E}}} in the following way. The Cartesian coordinates of a vector v of � →{\displaystyle {\overrightarrow {E}}} are the coefficients of v on the orthonormal basis � 1 , … , � � . {\displaystyle e_{1},\dots ,e_{n}.} For example, the Cartesian coordinates of a vector � v on an orthonormal basis ( � 1 , � 2 , � 3 ) {\displaystyle (e_{1},e_{2},e_{3})} (that may be named as ( � , � , � ) (x,y,z) as a convention) in a 3-dimensional Euclidean space is ( � 1 , � 2 , � 3 ) {\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})} if � = � 1 � 1 + � 2 � 2 + � 3 � 3 {\displaystyle v=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}. As the basis is orthonormal, the i-th coefficient � � \alpha _{i} is equal to the dot product � ⋅ � � . {\displaystyle v\cdot e_{i}.} The Cartesian coordinates of a point P of E are the Cartesian coordinates of the vector � � → . {\displaystyle {\overrightarrow {OP}}.} Other coordinates 3-dimensional skew coordinates Main article: Coordinate system As a Euclidean space is an affine space, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define affine coordinates, sometimes called skew coordinates for emphasizing that the basis vectors are not pairwise orthogonal. An affine basis of a Euclidean space of dimension n is a set of n + 1 points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point. Many other coordinates systems can be defined on a Euclidean space E of dimension n, in the following way. Let f be a homeomorphism (or, more often, a diffeomorphism) from a dense open subset of E to an open subset of � � . \mathbb{R} ^{n}. The coordinates of a point x of E are the components of f(x). The polar coordinate system (dimension 2) and the spherical and cylindrical coordinate systems (dimension 3) are defined this way. For points that are outside the domain of f, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the antimeridian, the longitude passes discontinuously from –180° to +180°. This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds. Isometries An isometry between two metric spaces is a bijection preserving the distance,[b] that is {\displaystyle d(f(x),f(y))=d(x,y).} In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm , {\displaystyle \|f(x)\|=\|x\|,} since the norm of a vector is its distance from the zero vector. It preserves also the inner product {\displaystyle f(x)\cdot f(y)=x\cdot y,} since {\displaystyle x\cdot y={\frac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).} An isometry of Euclidean vector spaces is a linear isomorphism.[c][8] An isometry � : � → � {\displaystyle f\colon E\to F} of Euclidean spaces defines an isometry � → : � → → � →{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if E and F are Euclidean spaces, O ∈ E, O′ ∈ F, and � → : � → → � →{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}} is an isometry, then the map � : � → � {\displaystyle f\colon E\to F} defined by � ( � ) = � ′ + � → ( � � → ) {\displaystyle f(P)=O'+{\overrightarrow {f}}\left({\overrightarrow {OP}}\right)} is an isometry of Euclidean spaces. It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces. Isometry with prototypical examples If E is a Euclidean space, its associated vector space � →{\displaystyle {\overrightarrow {E}}} can be considered as a Euclidean space. Every point O ∈ E defines an isometry of Euclidean spaces � ↦ � � → , {\displaystyle P\mapsto {\overrightarrow {OP}},} which maps O to the zero vector and has the identity as associated linear map. The inverse isometry is the map � ↦ � + � . {\displaystyle v\mapsto O+v.} A Euclidean frame ( � , � 1 , … , � � ) {\displaystyle (O,e_{1},\dots ,e_{n})} allows defining the map � → � � � ↦ ( � 1 ⋅ � � → , … , � � ⋅ � � → ) , {\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto \left(e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}\right),\end{aligned}}} which is an isometry of Euclidean spaces. The inverse isometry is � � → � ( � 1 … , � � ) ↦ ( � + � 1 � 1 + ⋯ + � � � � ) . {\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}} This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension. This justifies that many authors talk of � � \mathbb {R} ^{n} as the Euclidean space of dimension n. Euclidean group Main articles: Euclidean group and Rigid transformation An isometry from a Euclidean space onto itself is called Euclidean isometry, Euclidean transformation or rigid transformation. The rigid transformations of a Euclidean space form a group (under composition), called the Euclidean group and often denoted E(n) of ISO(n). The simplest Euclidean transformations are translations � → � + � Regards, Sam Latter Editor in Chief, Empire Financial Research P.S. When it comes to valuing this particular business, I don't think anyone else in the world outside this company knows as much about it as Whitney. I encourage you to watch his presentation and find out why he sees another "back up the truck" opportunity in it today. This move could help save your retirement, and you won't hear about it anywhere else... You can access his new presentation - and see how to access all of his other research, by [Whitney's message]( clicking here. [RelaxAndTrade]( From time to time, we send special emails or offers to readers who chose to opt-in. We hope you find them useful. 135 Auburn Ave NE Suite 201, Atlanta, GA 30303, United States To be sure our emails continue reaching your email box, please add our email address to your [whitelist](. [Privacy Policy]( | [Terms & Conditions]( | [Unsubscribe]( Copyright © 2023 Relax And Trade | All Rights Reserved