The Secret Life of the ax + b Group

Transcription

1 The Secret Life of the ax + b Group Linear function x ax + b are prominent if not ubiquitou in high chool mathematic, beginning in, or now before, Algebra I. In particular, they are prime exhibit in any dicuion of function. It i perhap trange, then, that the topic of compoition of linear function doe not receive more attention. Epecially, the aociated group, the affine group of the line, alo known a the ax + b group, along with it group law, i pretty univerally ignored. Thi note will explore ome way in which the ax + b group urface unacknowledged in high chool mathematic, and ome extenion of tandard topic which are uggeted by thi point of view. 1. Tranlation, dilation, and affine tranformation. Conider the function of tranlation by a contant a: T a : x x + a. Here a i any fixed number, and x i a real variable. A main property of function i that they can be compoed. We compute (T b T a )(x) = T b (T a (x)) = T a (x) + b = (x + a) + b = x + (a + b) = x + (b + a) = T b+a (x). (T rancomp) In other word, the compoite of T b and T a i T b+a. The family of tranlation i cloed under compoition, and for tranlation function, compoition mimic addition of real number. We may alo conider function of dilation, aka homogeneou linear function (or what in more modern mathematical parlance would be called imply linear function) D : x x. Here i a fixed non-zero real number, and x again i a real variable. If we compoe dilation, we get another one: (D t D )(x) = D t (D (x)) = td (x) = t(x) = (t)x = D t (x). (Dilcomp) Thu, dilation are another family of function which i cloed under compoition, and for dilation, compoition mimic multiplication of (non-zero number. We may alo compoe a tranlation and a dilation. Thi give a new kind of function, whoe traditional name in high chool algebra i linear function, but which we will try to call affine function, or affine mapping or affine tranformation. Thee are the mapping We can compoe two affine mapping: A,a (x) = T a D (x) = T a (D (x)) = x + a. (A,a A t.b )(x) = A,a (A t,b (x)) = A t,b (x) + a = (tx + b) + a = atx + b + a = A t,a+b (x). Thi how u that the collection of all affine mapping i cloed under compoition, and the compoition of affine tranformation i decribed by the rule (aka group law ) A,a A t,b = A t,a+b. (Affcomp) In thi connection, we note that A 1,0 i the identity map, and therefore that the compoition formula let u check that (A,a ) 1 = A 1. (AffInv) 1, a

2 Application 1: Solve the linear equation x + a = c. (L1) Solution: In term of affine mapping, thi equation ay that A,a (x) = c, o we can find x by inverting A,a : x = (A,a ) 1 (c) = A 1, a (c) = c a = c a. Remark: The equation x + a = c and a = c are often referred to in introductory algebra text a onetep equation, wherea the equation x + b = c i called a two-tep equation, reflecting the fact that we contruct A,a a a compoition of two tranformation, one dilation and one tranlation.. Conjugation and Fixed Point. Since the affine tranformation do not commute with each other, they act on themelve non-trivially by conjugation. Explicitly, we have In particular, A,a A t,b (A.,a ) 1 = A t,a+b A 1, a = A t,a+b ta = A t,()a+b. (Conj1) T a D t T a = A t,()a. (Conj) We are intereted in the fixed point of affine tranformation. Fixed point of tranformation behave in a very imple way under conjugation. Propoition.1: Let Φ and Ψ be tranformation of ome et X. Let F Ψ denote the et of fixed point for Ψ: F Ψ = {x X : Ψ(x) = x}. Then Thi i proved by a imple calculation. F Φ Ψ Φ 1 = Φ(F Ψ ). Of coure, the identity tranformation leave all point fixed. It i eay to check that non-trivial tranlation have no fixed point. Non-trivial dilation have the origin a a unique fixed point: if D (x) = x = x for any x 0, then ( 1)x = 0, whence 1 = 0, or = 1. Combining thi with the formula above for the conjugate of a dilation by a tranlation (or jut by direct computation), we conclude that Propoition.: If 1, the affine tranformation A,a ha the unique fixed point Corollary.3: Auming that 1 t, the affine tranformation A,a and A t.b commute with each other if and only if they have the ame fixed point. Proof: Two tranformation Φ and Ψ commute with each other if and only if the conjugate of Ψ by Φ i Ψ itelf. Applying thi to A,a and A t,b, and uing formula (Conj 1), we ee that A,a and A t,b commute a with each other if and only if (1 t)a + b = b, or (1 t)a = (1 )b, or 1 = b. But the two ide of the lat equation are exactly the fixed point of A,a and of A t,b repectively. Application : Solve the (general one-variable linear) equation a 1. x + a = tx + b. (L) Solution: In term of affine mapping, thi equation ay that A,a (x) = A t,b (x). Proceeding a before, and inverting A,a, we get x = (A,a ) 1 A t,b (x). Since the right hand ide of the equation alo involve x, we have not arrived at a olution. However, we have done omething. We can interpret thi equation a aying that x i a fiixed point for the tranformation (A,a ) 1 A t,b = A t = A t. According to Propoition., thi fixed point i x = b a 1 t = b a t., a + b, b a

3 Remark: Thi reult can of coure be found eaily by manipulation of the original equation. However, equation (L) i known to be coniderably harder for tudent to deal with than equation (L1). Equation (L1) can be olved by backtracking, or undoing the indicated operation in ucceion. In tranformational term, thi mean inverting A,a by firt applying (T a ) 1, and then applying (D ) 1. Thi trategy fail for equation (L); one mut add and ubtract term from both ide of the equation to iolate x on one ide, and one mut conolidate the term in x by uing the Ditributive Rule. The main point of the dicuion above i to point out the way in which equation (L) differ from (L1). Intead of calling for inverting a function, it call for finding a fixed point, a very different kind of problem. 3. Iteration and Difference Equation. It i often of interet to iterate tranformation. From the equation (TranComp) decribing the compoition of two tranlation, it i eay to conclude that for a tranlation T a, we have (T a ) n = T na. (T ranit) It i jut a eay to pa from the formula (Dilcomp) to an expreion for the iteration of a dilation: (D ) n = D n. (DilIt) Uing formula (Affcomp) to find the iteration of a general affine tranformation i not a traightforward. However, if we take adapt formula (Conj) by eting (1 t)a = b, then we can write A b D t (A b ) 1 = A t,b. Since conjugation preerve compoition, thi allow u to generalize formula (DilIt) to conclude that (A t,b ) n = A b (D t ) n (A b ) 1 = A tn, b(n ). (AffIt) Application 3: difference equation) a) Find a general formula for number {x n } defined by the iterative cheme (aka, x n+1 = x n + b. (T rdiff) b) Find a general formula for number {x n } defined by the iterative cheme x n+1 = tx n. (DilDiff) c) Find a general formula for number {x n } defined by the iterative cheme x n+1 = tx n + b. (AffDiff) Solution: Thee difference equation are repectively equivalent to x n+1 = T b (x n ), (T rdiff ) x n+1 = D t (x n ), (DilDiff ) and x n+1 = A t,b ((x n ). (AffDiff ) Conequently, they are olved by x n = (T b ) n (x o ) = x o + nb, x n = (D t ) n (x o ) = t n x o, 3 (T rdiffsoln) (DilDiffSoln)

4 and x n = (A t,b ) n (x o ) = A t n, b(n ) (x o ) = t n x o + b(1 tn ) = t n (x o b 1 t 1 t ) + b 1 t. (AffDiffSoln) Remark: i) Of coure, the third equation (AffDiff) include both of the other, o the correponding olution alo include both of the other. ii) Thee equation receive a fair amount of attention in ome algebra coure, and the difference cheme (TrDiff) i entailed in many pre- or proto-algebraic invetigation of pattern. iii) The olution (TrDiffSoln) and (DilDiffSoln) are relatively eay to come by. However, the general formula (AffDiff) preent more obtacle. Wherea the firt two application clearly are imply refined (ome might ay overly) interpretation of imple mathematic, the olution of (AffDiff) may be a place near where the line cro, and the tranformational olution i in ome way impler than a direct approach, which require the ummation of a geometric erie. Alternatively, thi approach can be thought of a howing that the ummation of a geometric erie, like o much ele in mathematic, i bound up with ymmetry. iv) The three difference cheme can be characterized a follow: a) A equence {x n } atifie an equation of the form (TrDiff) if and only if x n+1 x n = b (contant difference). b) A equence {x n } atifie an equation of the form (DilDiff) if and only if xn+1 x n = t (contant ratio). c) A equence {x n } atifie an equation of the form (AffDiff) if and only if xn+1 xn x n x n 1 = t (contant ratio of difference). 4. Action on Polynomial. So far, we have been looking at the compoition of affine function with themelve. However, we can compoe them with any function from R to R. We can compoe them on the right (precompoition) or on the left (potcompoition). The reult of thee operation how even more dramatically than the group law for the ax b group the difference order make when compoing function. Let P (x) = α n x n + α n 1 x n 1 + α n x n α 1 x + α o be a polynomial of degree n. Then and On the other hand and P T a (x) = α n (x + a) n + α n 1 (x + a) n 1 + α n (x + a) n α 1 (x + a) + α o, P D (x) = α n n x n + α n 1 n 1 x n 1 + α n n x n α 1 x + α o. T a P (x) = α n x n + α n 1 x n 1 + α n x n α 1 x + α o + a, D P (x) = α n x n + α n 1 x n 1 + α n x n α 1 x + α o. Thee formula are all expreed in term of power of x, except for the formula for P T a. We expand thi explicitly for quadratic and cubic polynomial. If Q(x) = α x + α 1 x + α o, then Q T a (x) = α x + (α 1 + α a)x + (α o + α 1 a + α a ). Similarly, for a cubic polynomial, C(x) = α 3 x 3 + α x + α 1 x + α o, then C T a (x) = α 3 x + (α + a)x + (α 1 + α a + a )x + (α 0 + α 1 a + α a + α 3 a 3 ). Given that we can tranform polynomial in thi explicit fahion by pre- and pot-compoing with affine tranformation, it i natural to ak whether we can ue tranformation to implify polynomial and polynomial equation. The anwer i a reounding ye! for quadratic polynomial, and ye alo for cubic. 4

5 A firt tep in implification of P i to eliminate the x n 1 term. Thi can in fact be done for a polynomial of any degree: we take a o = α n 1 nα n. Then we find that the x n 1 term alway vanihe. For a quadratic polynomial, we get Q(x) = Q T ao (x) = α x + (α o + α 1 ( α 1 α ) + α ( α 1 α ) ) = α x + (α o α 1 α + α 1 4α ) = α x + (α o α 1 4α ) = α x + ( 4α oα α 1 4α ), The implet thing to do next to implify Q i to pot-compoe with D 1 α. Thi give D 1 Q(x) = x + ( 4α oα α1 ) = x + γ α o. 4α Finally, if we potcompoe tranlation with γ o = 4αoα α 1, we obtain the monomial x. We record thi 4α reult. Propoition 4.1: Every quadratic polynomial Q i the tranform of Q o = x under precompoition by an appropriate tranlation, and potcompoition by an appropriate affine tranformation: Q = A,a Q o T b, for appropriate a, b and. Preciely, α x + α 1 x + α o = A,a x T b,, where = α, a = α 1 α, and b = α o a = 4α α o α 4α. Application 4: Solve the equation Q(x) = 0 for a given quadratic polynomial Q. Solution: If we have a quadratic equation, Q(x) = 0, then Propoition 3.1 ay that thi i equivalent to A,a Q 0 T b (x) = 0, with a, b and a pecified jut above. Tranforming by A,a 1, thi equation become Q o T a (x) = (A,a ) 1 (0) = a. Since Q o(x) = x, thi mean that T a (x) = ± a. Hence we finally conclude that x = T b (± a ) = b ± a. Remark: If we rewrite thi formula in term of the coefficient α j of Q, it become the quadratic formula. Thu, the quadratic formula may be thought of a a practical reflection of the fact that all quadratic function can be reduced to the implet poible one, Q o = x, by compoing appropriately with affine tranformation. In other word, if we know how to find quare root, we can olve any quadratic equation. Thi i another perpective on the quadratic formula: it reduce the olution of any quadratic equation to the problem of taking quare root. We turn now to cubic polynomial. If we tranlate by a o to eliminate the x term from C, we get and C(x) = C T ao (x) = α 3 x 3 + (α 1 + α ( α ) + ( α ) )x + (α o + α 1 ( α ) + α ( α ) + α 3 ( α ) 3 ) 5

6 = α 3 x 3 + (α 1 1 α )x + (α o α 1α + 3 α 3 7 α 3 α3 ) = β 3 x 3 + β 1 x + β o. If it hould happen that at thi point we have β 1 = 0, then we can proceed a we did with the quadratic, and olve c(x) = 0 by taking a cube root. From now on, we will aume that we are not in thi relatively imple cae, and that both β 1 and β 3 are non-zero. Thu we have two term involving x, the term x andx 3. We can adjut both of them by the ame calar via potcompoition with a dilation. Precompoition with a dilation adjut them by different power of the dilation factor. It turn out that we can adjut their ratio by any poitive factor. It turn out to be convenient to make β 1 three time the magnitude of β 3 Specifically, let u et = ± β 1 3β 3 1, where the ign i the ign of β 3, o that β 3 3 > 0. We then compute Q D (x) = β 3 3 x 3 + β 1 x + β o = 1 β 1 3 (x 3 ± 3x) + β 3 3β 3 1 o. Thi formula implie an analog of Propoition 3.1. Propoition 4.: Every cubic polynomial C i the tranform by pre- and potcompoition by affine tranform of one of three pecific cubic polynomial C ɛ (x) = x 3 + ɛ3x, where ɛ i equal to 1, 0, or -1: C(x) = A t,b C ɛ A,a, for appropriate affine tranformation A t,b and A,a. Preciely, α 3 x 3 + α x + α 1 x + α o = T,a (x 3 + ɛx) T t,b, where 3 t = α 3 δ, b = α 1 = α 3δ, = α 3 3 3δ t 3 = 9α3 In thee formula, ɛ = gn(3α 1 α 3 α ), and δ = 3α 1 α 3 α. Application 5: Solve the cubic equation C(x) = 0., a = α o + α3 9α 1 α α 3 7α3. Solution: Propoition 3. provide a reduction of thi problem to the problem of olving C ɛ (x) = y. More pecifically, if C i a in Propoition 3., then the equation C(x) = 0 i equivalent to A t,b C ɛ A,a (x) = 0, which i equivalent to C ɛ A,a (x) = A 1 t,b (0) = b t. If we can invert C ɛ, then we can turn thi equation to A,a (x) = C 1 ɛ ( b t ), or x = A 1,a(Cɛ 1 ( b t ) = 1 (C 1 ɛ ( b t ) a). If ɛ = 0, then computing Cɛ 1 amount to taking a cube root. Otherwie, there i no tandard notation which decribe c 1 ɛ. If ɛ = 1, then C ɛ i trictly monotone, and o i globally invertible a a function on R. Approximate value can be computed for Cɛ 1 (y) by mean of a computational cheme baed on the Newton-Raphon method. If ɛ = 1, then C ɛ i not globally invertible. For value of y between - and, there are three poible root x of C 1 (x) = y. One will be in the interval ( 1, 1), one will be greater than 1, and one will be le than -1. Iterative cheme to compute thee root preciely are fairly eay to contruct. Thi would be an approach to olving cubic equation parallel to the quadratic formula for quadratic. It i not nearly a elegant, becaue of the need for cae, but it i neverthele effective. 5. One parameter ubgroup. We hope the reader will agree that the application ketched above of the ax + b group in chool mathematic are intereting, and in fact, connect the ax + b group with key part (with the exception of normal form of cubic) of the high chool curriculum. However, the connection we will now dicu are in 6

7 ome ene more fundamental. They give inight into the nature of the function commonly tudied in high chool, and help to explain why we tudy thee particular function. The tranformation A,a depend on two parameter, the and the a. We are intereted in knowing the one-parameter ubgroup, the ubgroup which can be parametrized by a ingle variable. Formally, we call a et φ t of tranformation a one-parameter group if the φ t have multiplication law which mirror addition of number. That i a one-parameter group i a collection of tranformation φ uch that φ r φ = φ r+. OneP ar We hould alo require that φ depend continuouly on. We note that, a defined, a one-parameter group i not imply a collection of tranformation, it come with a parametrization φ. Thi mean that if we recale the group by ending c for ome contant c, then we will get the ame et of tranformation, but labeled by different numbe. We remark that uch recaling are the only way we can change parametrization, ubject to the conditiion (1P ). For if φ γ() i another parametrization of the T which give a one-parameter group, that i, which atifie (1P ), then γ() = c for ome contant c. To ee thi, we note that the group law implie that φ γ(r+) = φ γ(r) φ γ() = φ γ(r)+γ(). Auming that the φ are all ditinct, thi implie that γ( + t) = γ(r) + γ(). It i well-known, and we will not reheare the argument here, that the only continuou olution to thi functional equation are γ() = c for ome contant c. In the cae the parametrization of φ i not unique, the argument i lightly more elaborate. One mut analyze the poibilitie for redundancy. We will alo not do thi here. The one-to-one cae i the only one that will affect u here. An important fact about a one-parameter group i that all the operator in it necearily commute. Thi follow directly from the definition: the group law in a one-parameter group i modeled on the group law of R under addition, and thi i of coure commutative. Thi fact allow u almot immediately to determine the one-parameter ubgroup of the ax + b group; for we have een that if two (non-identity) tranformation commute, they are either both tranlation, or they have the ame fixed point (Corollary.3). Thi together with Propoition., the equation (Conj) of, and the dicuion above of reparametrization implie the following reult. Propoition 5.1: ax + b group are Up the conjugation and reparametrization, the one-parameter ubgroup of the T and D e 6. Function invariant under ymmetrie of both axe. We ak the following quetion: uppoe that we have a one-parameter group of affine tranformation acting on each axi: x φ r (x), and y ψ (y). What function Φ, if any, are compatible with thee action, in the ene that Φ(φ r (x)) = ψ r (Φ(x))? (Inv) We can think about the equation (Inv) in the following way. The graph of Φ i the curve coniting of the point [ ] x : x R Φ(x) in the carteian plane. We can combine the one-parameter group φ and ψ to get a one-parameter group of (affine) tranformation of the plane: [ x (φ ψ) r ( y ] ) = [ ] φr (x). (φ ψ) ψ r (y) The compatibility condition (Inv) imply amount to aking that the graph of Φ be tranformed into itelf by all tranformation (φ ψ) r, i.e., by the one-parameter group φ ψ. 7

8 What function are ingled out by the condition (Inv)? Evidently, thi depend on the one-parameter group φ and ψ. We know that, up to conjugation, there are two type of one parameter group, tranlation and dilation. Thi give u four cae to conider: We will conider thee cae in ucceion. TT) Both φ and ψ are group of tranlation. TD) The group φ conit of tranlation and the group ψ conit of dilation. DT) The group φ conit of dilation and the group ψ conit of tranlation. DD) Both group φ and ψ conit of dilation. Cae TT. Suppoe that φ (x) = x + a and ψ (y) = y + b. Then the condition (Inv) amount to Φ(x + a) = Φ(x) + b. Setting x = 0 and = t a, thi can be rewritten a Φ(t) = Φ(0) + b at. In other word, Φ i itelf jut an affine function (aka linear function, in the language of K-1), with dilation factor (aka lope) equal to the ratio b a of the caling of the x-tranlation group and the y-tranlation group. Cae TD. Suppoe again that φ (x) = x + a, but that now ψ x (y) = e b y. Then the condition (Inv) become Φ(x + a) = e b Φ(x). Again etting x = 0 and = t a, we find that Φ(t) = e b a t Φ(0). Thu in thi cae, Φ i a multiple of (or equivalently, a horizontal tranlation of) an exponential function. Cae DT. Here we revere the role of dilation and tranlation from the previou cae. We et φ (x) = e a x and ψ (y) = y + bt. Then the condition (Inv) read Φ(e a x) = Φ(x) + b. If we et x = 1 and e a = t, o that = ln(t) a, then we get the equation Φ(t) = Φ(1) + b a ln(t). Thu, Φ i an affine tranform of the natural logarithm function. The factor a b can be aborbed into the logarithm by uing a different bae, and the contant could alo be aborbed by tranlation Φ. Cae DD. Here both φ and ψ involve dilation: φ (x) = e a x, and ψ (y) = e b y. The compatibility condition (Inv) now read Φ(e a x) = e b Φ(x). Again etting x = 1 and e a = t, we ee that Φ(t) = e b Φ(1)) = (e a ) b a Φ(1) = t b a Φ(1). Thu, in thi cae, Φ i a multiple of a power function. In ummary, we ee that the baic function elementary function - linear function (aka affine or firt order function), exponential and logarithm function, and power function - which occupy o much attention in algebra/precalculu are ingled out by ymmetry conideration, pecifically, ymmetry with repect to affine tranformation. The other function which receive ubtantial attention in the late high chool curriculum are trigonometric function. Thee, epecially ine and coine, are alo deeply involved with ymmetry, pecifically with rotational ymmetry. Characteritic functional equation While it i debatable whether the above conideration need to be part of the high chool curriculum, I would argue that teacher hould undertand thee idea. Furthermore, there i another apect of thee idea which not only teacher hould be aware of, but hould be trongly conidered for incluion in normal precalculu coure. Thi i the matter of functional equation which are atified by, and which characterize, thee baic function. We will reprie the cae-by-cae dicuion above to bring out thee equation. Cae TT. Here the (Inv) relation i Φ(x + a) = Φ(x) + b. If we et a = x, then b = Φ(x ) Φ(0). Thu another wa of tating the relation (Inv) i Φ(x + x ) = Φ(x) + Φ(x ) Φ(0). (AffChar) We ee then that the function Φ = Φ Φ(0) which i the homogeneou linear function (aka, linear function) with the ame lope a Φ atifie the functional equation Φ(x + x ) = Φ(x) + Φ(x ). (LinChar)) A we have remarked, the equation (LinChar) characterize the function x cx: they are the only (continuou) function which atify (LinChar). Similarly the function x cx + d are characterized by the equation (AffChar). Indeed, a we have hown, Φ atifie (AffChar) if and only if Φ = Φ = Φ(0) atifie (LinChar). 8

9 Cae TD. The (Inv) relation i Φ(x + a) = e b Φ(x). Setting a = x, we have that e b = Φ(x ) Φ(0). Thu, in thi cae, Φ atifie the functional equation The correponding equation for the normalized function Φ = Φ(x + x ) = Φx)Φ(x ). (ExpM ultchar) Φ(0) Φ Φ(0) i Φ(x + x ) = Φ(x) Φ(x ). (ExpChar) Thi label capture the eential feature of the exponential function- that it convert addition into multiplication. A it label ugget, it characterize exponential function: any (continuou) function atifying it ha the form x c x for ome poitive number c (the bae of the exponent). Similarly, and by a reduction imilar to the one in cae TT, the equation (ExpMultChar) characterize multiple of exponential function. Cae DT. We will be omewhat briefer with the lat two cae. The (Inv) relation here i eentially the revere of the lat ituation: Φ(e a x) = Φ(x) + b. If we et e a = x, then b = Φ(x ) Φ(1), o we get the functional equation Φ(xx ) = Φ(x) + Φ(x ) Φ(1). (LogMultChar) The aociated normalized function i Φ = Φ Φ(1). It atifie the functional equation Φ(xx ) = Φ(x) + Φ(x ). (LogChar). Again the equation (LogCare) characterize logarithm function, and the equation (LogMultChar) characterize multiple of logarithm function. Cae DD. Finally, for the cae DD, the functional equation derived in the manner of the one above i The aociated homogeneou functional equation i Φ(xx ) = Φ(x)Φ(x ). (M ultp owerchar) Φ(1) Φ(xx ) = Φ(x) Φ(x ). (P owerchar) Again the only continuou olution to equation PowerChar are the exponential function. The equation (MultPowerChar) alo allow multiple of exponential function. We remark that, in the language of group theory, equation (PowerChar) i jut the functional equation for homomorphim from the multiplicative group of R to itelf. Thu, all the function that are invariant under one-parameter affine group which preerve the axe in the plane are characterized by imple functional equation. Thee equation are omewhat analogou to thoe characterizing the olution to the affine difference equation analyzed in 4. 9