Functions and mappings rFrom Wikipedia, the free encyclopediaContents1 2 2 real matrices 11.1 Prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equi-areal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Functions of 2 2 real matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 2 2 real matrices as complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Range (mathematics) 52.1 Distinguishing between the two uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Rayleigh dissipation function 73.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Reection (mathematics) 84.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Reection across a line in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Reection through a hyperplane in n dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Rigid transformation 135.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Distance formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Translations and linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Rotation of axes 16iii CONTENTS6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Examples in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4 Rotation of conic sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4.1 Identifying rotated conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.5 Generalization to several dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6 Examples in several dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6.1 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 216.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Chapter 12 2 real matricesIn mathematics, the set of 22 real matrices is denoted by M(2, R). Two matrices p and q in M(2, R) have a sum p+ q given by matrix addition. The product matrix p q is formed from the dot product of the rows and columns of itsfactors through matrix multiplication. Forq=_a bc d_,letq=_d bc a_.Then q q* = q* q = (ad bc) I, where I is the 22 identity matrix. The real number ad bc is called the determinantof q. When ad bc 0, q is an invertible matrix, and thenq1= q / (ad bc).The collection of all such invertible matrices constitutes the general linear group GL(2, R). In terms of abstractalgebra, M(2, R) with the associated addition and multiplication operations forms a ring, and GL(2, R) is its group ofunits. M(2, R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphicto the coquaternions, but has a dierent prole.The 22 real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesiancoordinate system into itself by the rule_xy__a bc d__xy_=_ax + bycx + dy_.1.1 ProleWithin M(2, R), the multiples by real numbers of the identity matrix I may be considered a real line. This real lineis the place where all commutative subrings come together:Let Pm = {xI + ym : x, y R} where m2 { I, 0, I }. Then Pm is a commutative subring and M(2, R) = Pmwhere the union is over all m such that m2 { I, 0, I }.To identify such m, rst square the generic matrix:_aa + bc ab + bdac + cd bc + dd_.12 CHAPTER 1. 2 2 REAL MATRICESWhen a + d = 0 this square is a diagonal matrix. Thus one assumes d = a when looking for m to form commutativesubrings. When mm= I, then bc = 1 aa, an equation describing a hyperbolic paraboloid in the space of parameters(a, b, c). Such an m serves as an imaginary unit. In this case Pm is isomorphic to the eld of (ordinary) complexnumbers.When mm = +I, m is an involutory matrix. Then bc = +1 aa, also giving a hyperbolic paraboloid. If a matrix is anidempotent matrix, it must lie in such a Pm and in this case Pm is isomorphic to the ring of split-complex numbers.The case of a nilpotent matrix, mm = 0, arises when only one of b or c is non-zero, and the commutative subring Pmis then a copy of the dual number plane.When M(2, R) is recongured with a change of basis, this prole changes to the prole of split-quaternions wherethe sets of square roots of I and I take a symmetrical shape as hyperboloids.1.2 Equi-areal mappingMain article: Equiareal mapFirst transform one dierential vector into another:_dudv_=_p rq s__dxdy_=_p dx + r dyq dx + s dy_.Areas are measured with densitydx dy , a dierential 2-form which involves the use of exterior algebra. Thetransformed density isdu dv= 0 + ps dx dy + qr dy dx + 0= (ps qr) dx dy= (det g) dx dy.Thus the equi-areal mappings are identied with SL(2, R) = {g M(2, R) : det(g) = 1}, the special linear group. Giventhe prole above, every such g lies in a commutative subring Pm representing a type of complex plane according tothe square of m. Since g g* = I, one of the following three alternatives occurs:mm = I and g is on a circle of Euclidean rotations; ormm = I and g is on an hyperbola of squeeze mappings; ormm = 0 and g is on a line of shear mappings.Writing about planar ane mapping, Rafael Artzy made a similar trichotomy of planar, linear mapping in his bookLinear Geometry (1965).1.3 Functions of 2 2 real matricesThe commutative subrings of M(2, R) determine the function theory; in particular the three types of subplanes havetheir own algebraic structures which set the value of algebraic expressions. Consideration of the square root functionand the logarithmfunction serves to illustrate the constraints implied by the special properties of each type of subplanePm described in the above prole. The concept of identity component of the group of units of Pm leads to the polardecomposition of elements of the group of units:If mm = I, then z = exp(m).If mm = 0, then z = exp(s m) or z = exp(s m).If mm = I, then z = exp(a m) or z = exp(a m) or z = m exp(a m) or z = m exp(a m).1.4. 2 2 REAL MATRICES AS COMPLEX NUMBERS 3In the rst case exp( m) = cos() + m sin(). In the case of the dual numbers exp(s m) = 1 + s m. Finally, in the caseof split complex numbers there are four components in the group of units. The identity component is parameterizedby and exp(a m) = cosh a + m sinh a.Now exp(am) = exp(am/2) regardless of the subplane Pm, but the argument of the function must betaken from the identity component of its group of units. Half the plane is lost in the case of the dual number structure;three-quarters of the plane must be excluded in the case of the split-complex number structure.Similarly, if exp(a m) is an element of the identity component of the group of units of a plane associated with 22matrix m, then the logarithm function results in a value log + a m. The domain of the logarithm function suers thesame constraints as does the square root function described above: half or three-quarters of Pm must be excluded inthe cases mm = 0 or mm = I.Further function theory can be seen in the article complex functions for the Cstructure, or in the article motor variablefor the split-complex structure.1.4 2 2 real matrices as complex numbersEvery 22 real matrix can be interpreted as one of three types of (generalized[1]) complex numbers: standard complexnumbers, dual numbers, and split-complex numbers. Above, the algebra of 22 matrices is proled as a union ofcomplex planes, all sharing the same real axis. These planes are presented as commutative subrings Pm. We candetermine to which complex plane a given 22 matrix belongs as follows and classify which kind of complex numberthat plane represents.Consider the 22 matrixz=_a bc d_.We seek the complex plane Pm containing z.As noted above, the square of the matrix z is diagonal when a + d = 0. The matrix z must be expressed as the sumof a multiple of the identity matrix I and a matrix in the hyperplane a + d = 0.Projecting z alternately onto thesesubspaces of R4yieldsz= xI + n, x =a + d2, n = z xI.Furthermore,n2= pI where p =(ad)24+ bc .Now z is one of three types of complex number:If p < 0, then it is an ordinary complex number:Let q= 1/p, m = qn . Then m2= I, z= xI + mp .If p = 0, then it is the dual number:z= xI + nIf p > 0, then z is a split-complex number:Let q= 1/p, m = qn . Then m2= +I, z= xI + mp .Similarly, a 22 matrix can also be expressed in polar coordinates with the caveat that there are two connectedcomponents of the group of units in the dual number plane, and four components in the split-complex number plane.4 CHAPTER 1. 2 2 REAL MATRICES1.5 References[1] Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine77(2):11829Rafael Artzy (1965) Linear Geometry, Chapter 2-6 Subgroups of the Plane Ane Group over the Real Field,p. 94, Addison-Wesley.Helmut Karzel & Gunter Kist (1985) Kinematic Algebras and their Geometries, found inRings and Geometry, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437509, esp 449,50, D.Reidel ISBN 90-277-2112-2 .Svetlana Katok (1992) Fuchsian groups, pp. 113, University of Chicago Press ISBN 0-226-42582-7 .Garret Sobczyk (2012). Chapter 2: Complex and Hyperbolic Numbers. New Foundations in Mathematics:The Geometric Concept of Number. Birkhuser. ISBN 978-0-8176-8384-9.Chapter 2Range (mathematics)For other uses, see Range.In mathematics, and more specically in naive set theory, the range of a function refers to either the codomain orXYf(x)f : X Yxf is a function from domain X to codomain Y. The smaller oval inside Y is the image of f . Sometimes range refers to the imageand sometimes to the codomain.the image of the function, depending upon usage. Modern usage almost always uses range to mean image.The codomain of a function is some arbitrary set. In real analysis, it is the real numbers. In complex analysis, it isthe complex numbers.The image of a function is the set of all outputs of the function. The image is always a subset of the codomain.56 CHAPTER 2. RANGE (MATHEMATICS)2.1 Distinguishing between the two usesAs the term range can have dierent meanings, it is considered a good practice to dene it the rst time it is usedin a textbook or article.Older books, when they use the word range, tend to use it to mean what is now called the codomain.[1][2] Moremodern books, if they use the word range at all, generally use it to mean what is now called the image.[3] To avoidany confusion, a number of modern books don't use the word range at all.[4]As an example of the two dierent usages, consider the function f(x) = x2as it is used in real analysis, that is, as afunction that inputs a real number and outputs its square. In this case, its codomain is the set of real numbers R , butits image is the set of non-negative real numbers R+, since x2is never negative if x is real. For this function, if weuse range to mean codomain, it refers to R . When we use range to mean image, it refers to R+.As an example where the range equals the codomain, consider the function f(x) = 2x , which inputs a real numberand outputs its double. For this function, the codomain and the image are the same (the function is a surjection), sothe word range is unambiguous; it is the set of all real numbers.2.2 Formal denitionWhen range is used to mean codomain, the range of a function must be specied. It is often assumed to be theset of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.When range is used to mean image, the range of a function f is {y | there exists an x in the domain of f such thaty = f(x)}. In this case, the codomain of f must be specied, but is often assumed to be the set of all real numbers.In both cases, image f range f codomain f, with at least one of the containments being equality.2.3 See alsoBijection, injection and surjectionCodomainImage (mathematics)Naive set theory2.4 Notes[1] Hungerford 1974, page 3.[2] Childs 1990, page 140.[3] Dummit and Foote 2004, page 2.[4] Rudin 1991, page 99.2.5 ReferencesChilds (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.).Springer. ISBN 978-0-387-74527-5. OCLC 173498962.Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.OCLC 52559229.Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics 73. Springer. ISBN 0-387-90518-9. OCLC 703268.Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0-07-054236-8.Chapter 3Rayleigh dissipation functionIn physics, the Rayleigh dissipation function, named for Lord Rayleigh, is a function used to handle the eects ofvelocity-proportional frictional forces in Lagrangian mechanics. It is dened for a system of N particles asF=12Ni=1(kxv2i,x + kyv2i,y + kzv2i,z).The force of friction is negative the velocity gradient of the dissipation function, Ff= vF . The function is halfthe rate at which energy is being dissipated by the system through friction.3.1 ReferencesGoldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. p. 24. ISBN0-201-02918-9.7Chapter 4Reection (mathematics)This article is about reection in geometry. For reexivity of binary relations, see reexive relation.In mathematics, areection (also spelledreexion)[1] is a mapping from a Euclidean space to itself that is anisometry with a hyperplane as a set of xed points; this set is called the axis (in dimension 2) or plane (in dimension3) of reection. The image of a gure by a reection is its mirror image in the axis or plane of reection. For examplethe mirror image of the small Latin letter p for a reection with respect to a vertical axis would look like q. Its imageby reection in a horizontal axis would look like b. A reection is an involution: when applied twice in succession,every point returns to its original location, and every geometrical object is restored to its original state.The term reection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely thenon-identity isometries that are involutions. Such isometries have a set of xed points (the mirror) that is an anesubspace, but is possibly smaller than a hyperplane. For instance a reection through a point is an involutive isometrywith just one xed point; the image of the letter p under it would look like a d. This operation is also known as a centralinversion (Coxeter 1969, 7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, thereection in the point situated at the origin is the same as vector negation. Other examples include reections in aline in three-dimensional space. Typically, however, unqualied use of the term reection means reection in ahyperplane.A gure which does not change upon undergoing a reection is said to have reectional symmetry.Some mathematicians use "ip" as a synonym for reection.[2]4.1 ConstructionIn plane (or 3-dimensional) geometry, to nd the reection of a point one drops a perpendicular from the point ontothe line (plane) used for reection, and continues it to the same distance on the other side. To nd the reection of agure, one reects each point in the gure.To reect point P in the line AB using compass and straightedge, proceed as follows (see gure):Step 1 (red): construct a circle with center at P and some xed radius r to create points A' and B' on the lineAB, which are equidistant from P.Step 2 (green): construct circles centered at A' and B' having radius r. P and Q will be the points of intersectionof these two circles.Point Q is then the reection of point P in line AB.4.2 PropertiesThe matrix for a reection is orthogonal with determinant 1 and eigenvalues (1, 1, 1, ... 1, 1). The product of twosuch matrices is a special orthogonal matrix which represents a rotation. Every rotation is the result of reecting in aneven number of reections in hyperplanes through the origin, and every improper rotation is the result of reecting84.3. REFLECTION ACROSS A LINE IN THE PLANE 9MC BAAB CM/2A reection through an axis followed by a reection across a second axis parallel to the rst one results in a total motion which is atranslation.in an odd number. Thus reections generate the orthogonal group, and this result is known as the CartanDieudonntheorem.Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reections in anehyperplanes.In general, a group generated by reections in ane hyperplanes is known as a reection group.Thenite groups generated in this way are examples of Coxeter groups.4.3 Reection across a line in the planeFor more details on reection of light rays, see Specular reection Direction of reection.Reection across a line through the origin in two dimensions can be described by the following formulaRefl(v) = 2v ll l l v10 CHAPTER 4. REFLECTION (MATHEMATICS)A BPA' B'QOPoint Q is reection of point P in the line ABWhere v denotes the vector being reected, l denotes any vector in the line being reected in, and vl denotes the dotproduct of v with l. Note the formula above can also be described asRefl(v) = 2Projl(v) vWhere the reection of line l on a is equal to 2 times the projection of v on line l minus v. Reections in a line havethe eigenvalues of 1, and 1.4.4 Reection through a hyperplane in n dimensionsGiven a vector a in Euclidean space Rn, the formula for the reection in the hyperplane through the origin, orthogonalto a, is given byRefa(v) = v 2v aa aawhere va denotes the dot product of v with a. Note that the second term in the above equation is just twice the vectorprojection of v onto a. One can easily check thatRefa(v) = v, if v is parallel to a, andRefa(v) = v, if v is perpendicular to a.4.4. REFLECTION THROUGH A HYPERPLANE IN N DIMENSIONS 11BCAA'B'C'/2A reection across an axis followed by a reection in a second axis not parallel to the rst one results in a total motion that is arotation around the point of intersection of the axes.Using the geometric product the formula is a little simplerRefa(v) = avaa2Since these reections are isometries of Euclidean space xing the origin they may be represented by orthogonalmatrices. The orthogonal matrix corresponding to the above reection is the matrix whose entries areRij= ij 2aiaja2where ij is the Kronecker delta.The formula for the reection in the ane hyperplane v a = c not through the origin isRefa,c(v) = v 2v a ca aa.12 CHAPTER 4. REFLECTION (MATHEMATICS)4.5 See alsoCoordinate rotations and reectionsHouseholder transformationInversive geometryPoint reectionPlane of rotationReection mappingReection groupSpecular reection4.6 Notes[1] Reexion is an archaic spelling.[2] Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p.251Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 64.7 ReferencesCoxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons,ISBN 978-0-471-50458-0, MR 123930Popov, V.L. (2001), Reection, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Reection, MathWorld.4.8 External linksReection in Line at cut-the-knotUnderstanding 2DReection and Understanding 3DReection by Roger Germundsson, The WolframDemon-strations Project.Chapter 5Rigid transformationIn mathematics, arigidtransformation (isometry) of a vector space preserves distances between every pair ofpoints.[1][2] Rigid transformations of the plane R2, space R3, or real n-dimensional space Rn are termed a Euclideantransformation because they form the basis of Euclidean geometry.[3]The rigid transformations include rotations, translations, reections, or their combination. Sometimes reectionsare excluded from the denition of a rigid transformation by imposing that the transformation also preserve thehandedness of gures in the Euclidean space (a reection would not preserve handedness; for instance, it wouldtransform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as properrigid transformations (informally, also known as roto-translations).In general, any proper rigid transformationcan be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as animproper rotation followed by a translation (or as a sequence of reections).Any object will keep the same shape and size after a proper rigid transformation.All rigid transformations are examples of ane transformations. The set of all (proper and improper) rigid transfor-mations is a group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of properrigid transformation is called special Euclidean group, denoted SE(n).In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to representthe linear and angular displacement of rigid bodies. According to Chasles theorem, every rigid transformation canbe expressed as a screw displacement.5.1 Formal denitionA rigid transformation is formally dened as a transformation that, when acting on any vector v, produces a trans-formed vector T(v) of the formT(v) = R v + twhere RT = R1 (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.A proper rigid transformation has, in addition,det(R) = 1which means that R does not produce a reection, and hence it represents a rotation (an orientation-preserving or-thogonal transformation).Indeed, when an orthogonal transformation matrix produces a reection, its determinantis 1.5.2 Distance formulaA measure of distance between points, or metric, is needed in order to conrm that a transformation is rigid. TheEuclidean distance formula for Rn is the generalization of the Pythagorean theorem. The formula gives the distance1314 CHAPTER 5. RIGID TRANSFORMATIONsquared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that isd(X, Y)2= (X1 Y1)2+ (X2 Y2)2+ . . . + (Xn Yn)2= (X Y) (X Y).where X=(X1, X2, ..., X) and Y=(Y1, Y2, ..., Y), and the dot denotes the scalar product.Using this distance formula, a rigid transformation g:RnRn has the property,d(g(X), g(Y))2= d(X, Y)2.5.3 Translations and linear transformationsA translation of a vector space adds a vector d to every vector in the space, which means it is the transformationg(v):vv+d. It is easy to show that this is a rigid transformation by computing,d(v + d, w + d)2= (v + d w d) (v + d w d) = (v w) (v w) = d(v, w)2.A linear transformation of a vector space, L:Rn Rn, has the property that the transformation of a vector, V=av+bw,is the sum of the transformations of its components, that is,L(V) = L(av + bw) = aL(v) + bL(w).Each linear transformation L can be formulated as a matrix operation, which means L:v[L]v, where [L] is an nxnmatrix.A linear transformation is a rigid transformation if it satises the condition,d([L]v, [L]w)2= d(v, w)2,that isd([L]v, [L]w)2= ([L]v [L]w) ([L]v [L]w) = ([L](v w)) ([L](v w)).Now use the fact that the scalar product of two vectors v.w can be written as the matrix operation vTw, where the Tdenotes the matrix transpose, we haved([L]v, [L]w)2= (v w)T[L]T[L](v w).Thus, the linear transformation L is rigid if its matrix satises the condition[L]T[L] = [I],where [I] is the identity matrix. Matrices that satisfy this condition are called orthogonal matrices. This conditionactually requires the columns of these matrices to be orthogonal unit vectors.Matrices that satisfy this condition form a mathematical group under the operation of matrix multiplication called theorthogonal group of nxn matrices and denoted O(n).Compute the determinant of the condition for an orthogonal matrix to obtaindet([L]T[L]) = det[L]2= det[I] = 1,5.4. REFERENCES 15which shows that the matrix [L] can have a determinant of either +1 or 1. Orthogonal matrices with determinant1 are reections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can beviewed as consisting of two manifolds in Rnxn separated by the set of singular matrices.The set of rotation matrices is called the special orthogonal group, and denoted SO(n). It is an example of a Lie groupbecause it has the structure of a manifold.5.4 References[1] O. Bottema & B. Roth (1990). Theoretical Kinematics. Dover Publications. reface. ISBN 0-486-66346-9.[2] J. M. McCarthy (2013). Introduction to Theoretical Kinematics. MDA Press. reface.[3] Galarza, Ana Irene Ramrez; Seade, Jos (2007), Introduction to classical geometries, BirkhauserChapter 6Rotation of axesAn xy-Cartesian coordinate system rotated through an angle to an x'y'-Cartesian coordinate systemIn mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to anx'y'-Cartesian coordinate systemin which the origin is kept xed and the x'- and y'-axes are obtained by rotating the x-and y-axes counterclockwise through an angle . A point P has coordinates (x, y) with respect to the original systemand coordinates (x', y') with respect to the new system.[1] In the new coordinate system, the point P will appear tohave been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more thantwo dimensions is dened similarly.[2][3] A rotation of axes is a linear map[4][5] and a rigid transformation.6.1 MotivationWhen we want to study the equations of curves and when we wish to use the methods of analytic geometry, coor-dinate systems become essential. When we use the method of coordinate geometry we place the axes at a positionconvenient with respect to the curve under consideration. For example, when we study the equations of ellipses andhyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin.But now suppose that we have a problem in which the curve (hyperbola, parabola, ellipse, etc.) is not situated so166.2. DERIVATION 17conveniently with respect to the axes. We would then like to change the coordinate system in order to have the curveat a convenient and familiar location and orientation. The process of making this change is called a transformationof coordinates.[6]The solutions to many problems can be simplied by rotating the coordinate axes to obtain new axes through the sameorigin.6.2 DerivationThe equations dening the transformation in two dimensions, which rotates the xy-axes counterclockwise through anangle into the x'y'-axes, are derived as follows.In the xy-system, let the point P have polar coordinates (r, ) . Then, in the x'y'-system, P will have polar coordinates(r, ) .We haveandSubstituting equations (1) and (2) into equations (3) and (4), we obtainEquations (5) and (6) can be represented in matrix form as_xy_=_ cos sin sin cos __xy_,which is the standard matrix equation of a rotation of axes in two dimensions.[8]The inverse transformation isor_xy_=_cos sin sin cos __xy_.18 CHAPTER 6. ROTATION OF AXES6.3 Examples in two dimensions6.3.1 Example 1Find the coordinates of the point P1= (x, y) = (3, 1) after the axes have been rotated through the angle 1= /6, or 30.Solution:x=3 cos(/6) + 1 sin(/6) = (3)(3/2) + (1)(1/2) = 2y= 1 cos(/6) 3 sin(/6) = (1)(3/2) (3)(1/2) = 0.The axes have been rotated counterclockwise through an angle of1=/6 and the new coordinates areP1=(x, y) = (2, 0) . Note that the point appears to have been rotated clockwise through /6 , that is, it now coincideswith the (new) x'-axis.6.3.2 Example 2Find the coordinates of the point P2= (x, y) = (7, 7) after the axes have been rotated clockwise 90, that is, throughthe angle 2= /2 , or 90.Solution:_xy_=_ cos(/2) sin(/2)sin(/2) cos(/2)__77_=_0 11 0__77_=_77_.The axes have been rotated through an angle of2= /2 , which is in the clockwise direction and the newcoordinates are P2= (x, y) = (7, 7) . Again, note that the point appears to have been rotated counterclockwisethrough /2 .6.4 Rotation of conic sectionsMain article: Conic sectionThe most general equation of the second degree has the formThrough a change of coordinates (a rotation of axes and a translation of axes), equation (9) can be put into a standardform, which is usually easier to work with. It is always possible to rotate the coordinates in such a way that in the newsystem there is no x'y' term. Substituting equations (7) and (8) into equation (9), we obtainwhereIf we select so that cot 2 = (AC)/B we will have B= 0 and the x'y' term in equation (10) will vanish.[11]When a problem arises with B, D and E all dierent from zero, we may eliminate them by performing in successiona rotation (eliminating B) and a translation (eliminating the D and E terms).[12]6.5. GENERALIZATION TO SEVERAL DIMENSIONS 196.4.1 Identifying rotated conic sectionsA non-degenerate conic section given by equation (9) can be identied by evaluating B24AC . The conic sectionis:___an ellipse or a circle, if B24AC0.[13]6.5 Generalization to several dimensionsSuppose a rectangular xyz-coordinate system is rotated around its z-axis counterclockwise (looking down the positivez-axis) through an angle , that is, the positive x-axis is rotated immediately into the positive y-axis. The z-coordinateof each point is unchanged and the x- and y-coordinates transform as above. The old coordinates (x, y, z) of a pointQ are related to its new coordinates (x', y', z') by__xyz__=__cos sin 0sin cos 00 0 1____xyz__. [14]Generalizing to any nite number of dimensions, a rotation matrix A is an orthogonal matrix that diers from theidentity matrix in at most four elements. These four elements are of the formaii= ajj= cos and aij= aji= sin ,for some and some i j.[15]6.6 Examples in several dimensions6.6.1 Example 3Find the coordinates of the point P3= (w, x, y, z) = (1, 1, 1, 1) after the positive w-axis has been rotated throughthe angle 3= /12 , or 15, into the positive z-axis.Solution:____wxyz____=____cos(/12) 0 0 sin(/12)0 1 0 00 0 1 0sin(/12) 0 0 cos(/12)________wxyz________0.96593 0.0 0.0 0.258820.0 1.0 0.0 0.00.0 0.0 1.0 0.00.25882 0.0 0.0 0.96593________1.01.01.01.0____=____1.224751.000001.000000.70711____.6.7 See alsoRotation20 CHAPTER 6. ROTATION OF AXES6.8 Notes[1] Protter & Morrey (1970, p. 320)[2] Anton (1987, p. 231)[3] Burden & Faires (1993, p. 532)[4] Anton (1987, p. 247)[5] Beauregard & Fraleigh (1973, p. 266)[6] Protter & Morrey (1970, pp. 314-315)[7] Protter & Morrey (1970, pp. 320-321)[8] Anton (1987, p. 230)[9] Protter & Morrey (1970, p. 320)[10] Protter & Morrey (1970, p. 316)[11] Protter & Morrey (1970, pp. 321-322)[12] Protter & Morrey (1970, p. 324)[13] Protter & Morrey (1970, p. 326)[14] Anton (1987, p. 231)[15] Burden & Faires (1993, p. 532)6.9 ReferencesAnton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduc-tion to Groups, Rings, and Fields, Boston: Houghton Miin Co., ISBN 0-395-14017-XBurden, Richard L.; Faires, J. Douglas (1993),Numerical Analysis (5th ed.), Boston: Prindle, Weber andSchmidt, ISBN 0-534-93219-3Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading:Addison-Wesley, LCCN 760870426.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 216.10 Text and image sources, contributors, and licenses6.10.1 Text 2 2 real matrices Source: https://en.wikipedia.org/wiki/2_%C3%97_2_real_matrices?oldid=660100486 Contributors: Gareth Owen,Toby Bartels, Michael Hardy, TakuyaMurata, Giftlite, BenFrantzDale, Rgdboer, Jheald, SmackBot, Incnis Mrsi, Lambiam, Jim.belk,STBot, It Is Me Here, Haseldon, DavidCBryant, Geometry guy, Anchor Link Bot, Sun Creator, Marc van Leeuwen, TutterMouse, Yobot,AnomieBOT, BG19bot, Mark L MacDonald, Loraof and Anonymous: 12 Range (mathematics) Source: https://en.wikipedia.org/wiki/Range_(mathematics)?oldid=675918771 Contributors: Zundark, Amillar,Michael Hardy, Ellywa, Glenn, Andres, Hectorthebat, Charles Matthews, Dysprosia, Quux, Haukurth, Hyacinth, Frazzydee, Bearcat,Robbot, Sander123, Tobias Bergemann, Mor~enwiki, Giftlite, BenFrantzDale, Lethe, Jackol, Utcursch, Profvk, Abdull, Flyhighplato,Vivacissamamente, PhotoBox, Rcog, Paul August, Shanes, Bobo192, MPerel, Dallashan~enwiki, Arthena, Lord Pistachio, Oleg Alexan-drov, Ron Ritzman, Joriki, BD2412, Dpr, Rjwilmsi, Aurochs, Salix alba, DonSiano, Miserlou, FlaBot, VKokielov, RexNL, Gurch,Algebraist, RussBot, Bhny, Rick Norwood, Jonathan Webley, Welsh, Trovatore, Gareth Jones, Bota47, Closedmouth, JahJah, Spliy,Allens, SmackBot, Andy M. Wang, PrimeHunter, RayAYang, Can't sleep, clown will eat me, Radagast83, Daqu, Nakon, DMacks, Vildri-cianus, ArglebargleIV, Giovanni33, Eriatarka, Heimstern, JorisvS, Jim.belk, KHAAAAAAAAAAN, Mauro Bieg, Fan-1967, Irides-cent, Ouishoebean, JForget, CBM, Only2sea, Gregbard, Gogo Dodo, WISo, A3RO, JAnDbot, Gcm, Thenub314, Hut 8.5, PhilKnight,K61824, Penubag, Usien6, JoergenB, JaGa, MartinBot, Manticore, J.delanoy, Mange01, AntiSpamBot, Quantling, RB972, VolkovBot,LokiClock, Boute, Oshwah, Anonymous Dissident, Finlux, JhsBot, LeaveSleaves, Dmcq, SieBot, Captain Yankee, Paolo.dL, ClueBot,The Thing That Should Not Be, JP.Martin-Flatin, Arunsingh16, NClement, Excirial, Subdolous, Ottawa4ever, Crazy Boris with a redbeard, Guarracino, Alexius08, Ejosse1, Xvijayx, Addbot, Willking1979, LaaknorBot, Super Spider, Playclever, AnomieBOT, JackieBot,9258fahskh917fas, Ulric1313, Flewis, ArthurBot, Bdmy, The Banner, Raamaiden, WaysToEscape, SD5, Erik9bot, Tbunke, Sikevux,Haein45, Pinethicket, Retired user 0001, Vrenator, OlegGerdiy, Minimac, DARTH SIDIOUS 2, Jowa fan, Adam10101, Tommy2010,Osure, DASHBotAV, ClueBot NG, Bped1985, Widr, Mal48, Juro2351, Cispyre, Stephenwanjau, Solomon7968, DPL bot, Anbu121,Zatrp, Brirush, Loraof and Anonymous: 181 Rayleigh dissipation function Source: https://en.wikipedia.org/wiki/Rayleigh_dissipation_function?oldid=588359826 Contributors: Ja-son Quinn, Alvin Seville and Antiqueight Reection(mathematics) Source: https://en.wikipedia.org/wiki/Reflection_(mathematics)?oldid=664834535 Contributors: The Anome,SimonP, Patrick, Mdupont, TakuyaMurata, Glenn, AugPi, Charles Matthews, Dysprosia, Phys, MathMartin, Tosha, Giftlite, Frop-u, Tomruen, PhotoBox, Rich Farmbrough, Paul August, Gonzalo Diethelm, Zaslav, BenjBot, Rgdboer, Jet57, MiguelTremblay, OlegAlexandrov, Joe Decker, 25~enwiki, Nneonneo, Yamamoto Ichiro, Eubot, Alexb@cut-the-knot.com, Mathbot, Gurch, Scythe33, Chobot,YurikBot, Wavelength, Shell Kinney, Gwaihir, Grafen, Phgao, MathsIsFun, Mhss, Tamfang, Mecrazywong, Goodnightmush, Jim.belk,Cydebot, Thijs!bot, Headbomb, Oemb1905, Salgueiro~enwiki, .anacondabot, David Eppstein, J.delanoy, Wandering Ghost, Comet-styles, RJASE1, Pleasantville, JohnBlackburne, LokiClock, TXiKiBoT, Ceranthor, Rknasc, Cryonic07, SieBot, Jsc83, Paolo.dL, DanDs,Aboluay, ClueBot, GorillaWarfare, Marino-slo, Jan1nad, Razorame, Marc van Leeuwen, Addbot, Fgnievinski, Charcole125, Laaknor-Bot, Dayewalker, Luckas-bot, JackieBot, Loveless, Dako1, Sawomir Biay, Tkuvho, DASHBot, EmausBot, Acather96, ScottyBerg,Wikipelli, K6ka, Slawekb, ClueBot NG, Wcherowi, Brad7777, Fiboman, CopperSolder208, WillemienH and Anonymous: 57 Rigidtransformation Source: https://en.wikipedia.org/wiki/Rigid_transformation?oldid=649127814 Contributors: Edward, MichaelHardy, Anders Feder, Rgdboer, Jheald, Malcolma, SmackBot, Jab843, Khazar, JamesBWatson, R'n'B, Morenooso, Paolo.dL, CorenSearch-Bot, Staticshakedown, Yobot, Sawomir Biay, OrenSchwartz, Prof McCarthy, RudolfRed and Anonymous: 5 Rotationofaxes Source: https://en.wikipedia.org/wiki/Rotation_of_axes?oldid=673814734Contributors: Longhair, Gene Nygaard,Mindmatrix, Regre7, SDC, Joe Decker, SmackBot, Dr.enh, Alaibot, MER-C, InkKznr, Mchoi815, Leonard^Bloom, Materialscientist,J04n, GoingBatty, Anita5192, Climaxcraftt, Kalyan282 and Anonymous: 146.10.2 Images File:Codomain2.SVG Source: https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG License: Public domain Con-tributors: Transferred from en.wikipediaOriginal artist: Damien Karras (talk). Original uploader was Damien Karras at en.wikipedia File:Perpendicular-construction.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Perpendicular-construction.svgLicense: CC-BY-SA-3.0 Contributors: ? Original artist: ? File:Rotation_of_coordinates.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9f/Rotation_of_coordinates.svg License:CC BY-SA 3.0 Contributors: Own work Original artist: Guy vandegrift File:Science.jpg Source: https://upload.wikimedia.org/wikipedia/commons/5/54/Science.jpg License: Public domain Contributors: ?Original artist: ? File:Simx2=rotOK.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/da/Simx2%3DrotOK.svg License: CCBY2.5 Con-tributors: File:Simx2=rotOK.png Original artist: Simx2=rotOK.png: Toobaz File:Simx2=transl_OK.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/67/Simx2%3Dtransl_OK.svg License: CCBY-SA4.0 Contributors: Original artist: Fred the Oyster6.10.3 Content license Creative Commons Attribution-Share Alike 3.0