Introduction to Abstract Algebra
Introduction to Abstract Algebra MATH 113
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Date Created: 10/22/15
MATH 54 Lecture Notes 1 G81 Carter August 4 2006 1 Abel s Theorem Consider a linear homogeneous differential equation y p1ty 1 pnty 0 1 where each Pit is continuous on an open interval I containing a point to Let this equation have solutions y1y2yn These solutions may or may not be linearly independent We wish to find a formula for Wy1y2 yn which depends only on 1171tpgt7 7pnt as much as possible First we need to try to compute W We will use the following formula for the determinant detA Z 10a1a1a2a2 39an0n 065 where A is an n X n matrix with entries a The a 6 Sn and 71 parts are probably nonsensel to you7 but don t worry7 because they re not that important The point of the above formula is that the determinant can be written as a sum7 where each term in the sum is a product of n entries in the matrix7 each from a different row and different column We will also use the following generalized product rule fle39quotfnyfif2quot39fnflf quot39fnquot39f1f2quot39fp Using these two formulas7 we obtain a nil W Z 1 lt9a1yiy2quot39yun gt 065 Z 40 y71yz72 39 39 5231 901Wir2 39 39 5231 39 39 39 WOW72 39 39 yes Z UUyzrawzra 39 39 3231 Z UUyaquya 39 39 3231 39 39 39 Z UUyaquya 39 39 yes yes yes yl y y y1 y2 yn y1 y2 yn yl y y yl y y yl y y a 2 quot39 2 71 71 71 71 71 71 yln y n y yln y n y yln y n y y1 y2 yn yl y y yln y n y 1This footnote is only present for completeness sake and in case anyone who plans on being a math major wants to read it Here Sn refers to the set of all permutations of the set 12 n Each 7i in the subscript denotes the ieth element in that permutation The quantity 71 denotes the sign of the permutation It is 1 for permutations that are the product of an even number of transpositions and 71 for permutations that are the product of an odd number of transpositions This topic is covered in MATH 113 Using row reduction operations which preserve the determinant and the fact that each y is a solution to 1 we obtain 91 M yn yl y y W a 1 Pl V lt01 yin l Pl y ni 1 i101 y il Then P1tW 0 This equation is separable so we have 710175 dt Therefore Wt ce 0 1711 d 2 for some 5 which is constant with respect to t Equation 2 is called Abel s Theorern It is Theorem 332 and t Exercise 4120 in Boyce DiPrirna Since e7 fto 1711 11 y 0 for all t E I and c does not depend on it this implies that Wt is either zero for all t E I or nonzero for all t E I To summarize if y1y2 yn are solutions to 1 then the following are equivalent 1 The functions yl yg yn span the solution set of 12 2 The functions yl yg yn are linearly equivalent over I 3 The Wronskian Wy1y2 yn t y 0 for some t E I 4 The Wronskian Wy1y2 ynt y 0 for all t E I Exercise 3322 A linearly independent set of solutions must have a nonzero Wronskian This implies that pt 0 for all t E I Exercise 3324 The Wronskian Wy1y2 would have to be zero at such a point so it would be zero over all of Exercise 3328 On 01 f 9 On 710 f 9 However on 711 consider the linear rnap h12 Th7lth712 The map T sends f and g to a basis for R2 so they are linearly independent However their Wronskian is zero 2 Systems of Linear FirstOrder Differential Equations 21 Existence and Uniqueness Let I be an open interval containing a point to Let At be an n X n rnatrix where each entry is a function of t which is continuous on I Also let gt be an n elernent vector where each entry is also a function of t which is continuous on I Then a matrix equation X O Atxt g0 3 is what we call a system of linear first order differential equations Here taking the derivative of a vector simply means taking the derivative of each entry A solution to 3 is an n elernent vector xt where each entry is a differentiable function oft and 3 holds for all t E I Under these conditions for any x0 6 CC there exists a unique solution to 3 such that xt0 x0 This is Theorem 712 in Boyce DiPrirna We can rewrite 3 as x c 7 Alttgtxlttgt go This way the left hand side of the equation is a linear transformation of xt Then as with a single differential equation the corresponding hornogeneous system of equations is X O 7 Atxt 0 4 and the set of all solutions to 3 is obtained by taking any particular solution and then adding all solutions to 2BoyceDiFrirna refers to this as 111112 yn being a fundamental set of solutions for 22 HigherOrder Linear Differential Equations Suppose we have a linear differential equation W p1ty 1 pnty 9W We can turn this into a system of first order linear differential equations by introducing new variables For each 1 g 239 lt n7 let ui y i Then we have the following first order linear differential equations 9 U1 ul ug 12 ug 71272 unil 71271 905 P1tun71 p2tun72 39 39 39 Pnty If we let y 0 ul 0 X 7 2 7g 0 unil 905 and 0 1 0 o 0 0 1 o A 0 0 0 0 7PM pnidt 7mm 71010 then the above system becomes x Ax go For an example of this7 take the differential equation y ay by 0 Then we will set u y 7 so that y u and u 7au 7 by is our system of differential equations This is equivalent to y 7 0 1 y u 7 7b 7a u i The characteristic polynomial of this matrix is A71 7 2 b Aa 7A aAb which is the same characteristic equation we had been using for the second order equation