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Further Mathematics Revision GROUP THEORY Closure: no new elements created Identity: element that leaves other elements unchanged: e Inverse element: element, which if applied to another element creates the identity Self-inverses have order 2 a2=e Associativity: a ж (b ж c) = (a ж b) ж c Commutative: a ж b = b ж a; Latin square property holds Finite group: number of elements is not infinite. Order of a group: number of elements Order of an element: how many times must be combined with itself to make e Element a, order of group n an=e Generator: element a, that powers of a make up all the members of the group If order of group is prime, all elements are generators. Cyclic group: has at least one generator Klein group: group order 4, all self-inverses Prime order groups: no sub-groups, all elements generators, cyclic group Lagrange: order of subgroup: factor of group order Isomorphic: same as…; a=x, b=y,…; Operator: binary operation Cancellation laws: ab=ad b=d (a-1 a)b=(a-1 a)d eb=ed b=d QED -1 -1 fg=hg f=h f(g g)=h(g g) fe=he f=h QED RINGS Two rules of combination: - Under addition –> forms a group (commutative) - Under multiplication -> closed and associative (may not have identity) Commutative ring: if multiplication is commutative Distributive laws apply a(b+c) = ab + ac (b+c)a = ba + ca Ring with unity: has a multiplication identity -> each element has en inverse FIELD: Ring whose non-zero element forms a group under multiplication. Further Mathematics Revision PROOF Proof by contradiction: assume the opposite and prove wrong √3 is irrational Assume √3 is rational √3= a/b a, b have no common factor 3= a²/b² 3b²=a² a² = 3m a=3m … 3b²=(3m)² 3b²=9m² b²=3m² b² = 3m b=3m Contradiction a and b have a common factor √3 cannot be rational. √2 is irrational Assume √2 is rational. It can then be expressed as a fraction in it’s lowest form. √2=p/q p, q have no common factor 2=p2/q2 2q2=p2 p2 is even p is also even p=2k … 2q2=4k2 q2=2k2 q2 is even q is also even q=2m √2=2k/2m p and q have common factors. Contradiction √2 cannot be rational √2 is irrational Direct Proof/Algebraic Proof Sum of 3 consecutive integers is a multiple of 3 n, n+1, n+2 n + n + 1 + n + 2 = 3n + 3 = 3(n +1) = 3m, Sigma: sum of… 5r=1∑r = 1 + 2 +3 +4 +5 4r=1∑r(r-1)= 0 +2 +6 +12 mεZ 5r=1∑r2= nr=1∑r= 1 +4 +9 +16 +25 1 +2 +3 +4 +…+n …Prove that if n2=even then n=even too. even= 2m show even number squared is multiple of 2 2m=(n+1)2 n is odd 2 2 2m=n +2n+1=odd + even + odd= odd + even + odd=even QED Proof by induction: - Show true for n=1 - Assume true for n=k - Show is true for n=k+1 (do not assume!) - True for K implies truth for k+1 by induction true for all k Further Mathematics Revision SEQUENCES Convergent: sequence that has a limit Divergent: sequence with no limit Periodic sequence: sequence with pattern Arithmetic sequences: Un=a +(n-1)d if Un – Un-1 is a constant Geometric sequences: Un=arn-1 if Un/Un-1 is a constant eg Un+1=4Un ->Un=a4n-1 Difference equations: General solution: Particular solution: n-1 n-1 Geometric: Un=ar (=a4 ) Un=5.4n-1 Arithmetic: Un=a+d (=3n+(n-3)) Un=3n +3 Categorisation of difference equations Difference equation: recurrence equation Order: number of previous terms needed to find given term An+1=(2n +3)an order 1 Linearity: if each term mentioned is to the power of 1 An+1= n2an – 3an-1 linear Constant coefficients: if all coefficients are constant An+1= 3an – 5an-1 +n constant coefficients An+1= nan + 3 not constant coefficients Homogeneity: if equation contains at least 1 previous term An+1 = 5an – n2√an-1 homogenous; a to the power is not valid. An+1 = 2an2 not homogenous Monotonic sequences: an increasing or decreasing sequence; If Un+1 – Un > 0 increasing sequence If Un+1 – Un < 0 decreasing sequence n +1 n +2 n (n + 1 )2 n(n + 2 ) n 2 + 2n + 1 n 2 = = n +1 (n+1)(n+2) (n+1)(n+2) 2n 1 = (n+1)(n+2) n+1, n+2, 1 are positive. Positive/positive is positive. increasing monotonic LIMITS: divide by dominant terms + factorise lim(Un)= L1 lim(Vn)=L2 L1, L2 need to exist lim(Un + Vn)= L1+ L2 lim(Un x Vn)= L1x L2 p lim(1/n )= 0 lim(rn)=0 if –1<r<1 n2 + 1 n 2 + 2n 2n lim = 3 3.2 n = n2 2 n +1 3 = 1 3 lim(Un/Vn)= L1/L2 Further Mathematics Revision COMPLEX NUMBERS √-8 = (√8)i De Moivre’s Theorem: (cosθ +isinθ)n= cosnθ +isinnθ cos(A+B)= cosAcosB – sinAsinB sin(A+B)= sinAcosB + cosAsinB cosθ= cos(-θ) sinθ= -sin(-θ) Complex polynomials use formula (–b+-√(b²-4ac))/2a Real coefficients answers in conjugate pairs Examples: (a) Find (√3 + i)5 using De Moivre θ= п/6 r=2 z=√3 + i= 2(cosп/6 +isinп/6) z5=25(cosп/6+isinп/6)5 =32(cos5п/6 +isin5п/6)=32(-√3/2+i/2)=-16√3+16i (b) Find z5= 3+3i θ=п/4 r=√18 z 5 = 18 cos 5 z 5 =(r 1 ) (cos 5 r= p p + isin 4 4 + isin ) = ; z=r 1 (cosq + isin q ) 18 cos 4 + isin 4 2pp p 2pp p p 18 =10 18 ; 5 q = ; q = + = + 4 5ґ4 5 20 5 5solutionswhen q = p 9 p 17 p 25 p 33 p , , , , 20 20 20 20 20 Exponential form: eiθ=cosθ +isinθ z= r(cosθ + isinθ) = reiθ e2-3i = e²e-3i = e²(cos(-3) + isin(-3)) 2=eln2 2i=(eln2)i=eiln2= cosln2 + isinln2 ax+iy=axaiy=ax(cos(ylna) + isin(ylna)) Further Mathematics Revision VECTOR SPACES Closure: - Addition: u+v is in V - Scalar multiplication: cu is in V, for all u, c Identity: - Addition: there exists an element 0 in V such that u+0=u - Scalar multiplication: 1u=u Associativity: - Addition: (u+v)+w=u+(v+w) - Scalar multiplication: c(du)=(cd)u Distributivity: - Distributive 1: c(u+v)=cu+cv - Distributive 2: (c+d)u=cu+du Commutativity under addition: u+v=v+u Inverses under addition: u+ -u=0 Subspaces: subset of a vector space (must contain zero vector) - Closed under addition and scalar multiplication - Subspace inherits properties from vector space Basis: set of vector, when combined can generate all other vectors in V; Very efficient, e.g. M22 n(S)=4 1 0 0 0 , 1 , 0 Standard basis: e.g. V3= i, j, k: 0 0 1 Spanning Sets: basis, but less efficient Span: set of all linear combinations Dimension: number of elements in a basis of that vector space Summative results: let V be a vector space with dimV=n, then: - Any linear independent set in V contains at most n vectors - Any spanning set for V contains at least n vectors - Any linearly independent set of exactly n vectors in V is a basis for V - Any spanning set for V of exactly n vectors is a basis for V - Any linearly independent set in V can be extended to a basis for V - Any spanning set for V can be reduced to a basis for V Linear dependent: if there are scalars (c1c2c3…) such that c1v1+c2v2+…=0 -If at least one of its elements can be written as a combination of the others Linear independent: not linear dependent Isomorphism of vector spaces: all vectors are isomorphic Further Mathematics Revision MATRICES Matrix: rectangular array of numbers, not commutative under multiplication A matrix is defined to be the number of rows x number of columns 1 0 0 9 3 by 2 order of a matrix 8 4 1xn: row matrix nx1: column matrix nxn: square matrix Element: each number in the matrix Adding and subtracting: can only be done with matrices of the same order. corresponding elements Multiplying matrices: associative but not commutative 1 0 1ґ0 + 0ґ2 1ґ1 + 0ґ5 1ґ1 + 0ґ6 0 1 1 0 1 1 0 9 ґ = 0 ґ 0 + 9 ґ 2 0 ґ 1 + 9 ґ 5 0 ґ 1 + 9 ґ 6 = 18 45 54 2 5 6 8 4 8ґ0 + 4ґ2 8ґ1 + 4ґ5 8ґ1 + 4ґ6 8 28 32 nxm by mxp nxp, 2x4 by 4x3 2x3 1 0 0 Identity matrix: I, only exist for square matrices. 1 0 , 0 1 0 Multiplication by identity is commutative: IA=AI=A 0 1 0 0 1 Inverse of a matrix: inverse p q r s ® s r q s determinant ґ ® p r 1 q s ® p ps qr r q p Singular matrix: one that has no inverse the determinant is 0 (ps-qr=0) M22 forms a vector space under normal matrix addition and scalar multiplication