Math 122, Midterm CheatSheet

Peter Chang

October 16, 2015

1 Groups

1.1 Laws of Composition

• Law of Composition is a map:S × S → S

• Associative law is more fundamental because function compositions are associative.

• Identity for a law of composition: e ∈ S such that

ea = ae = a

• An element a ∈ S is invertible if ∃b ∈ S such that:

ab = ba = 1

1.2 Groups and Subgroups

• Group:(ab)c = a(bc)

1 ∈ G∀a ∈ G ∃a−1 ∈ G

• Abelian group is a group whose law of composition is commutative.

• The order of a group G:|G| = number of elements of G

• Cancellation Law:ab = ac or ba = ca→ b = c

• General Linear Group:GLn = {n× n invertible matrices A}

• Permutation: if M is a map from T to T , the invertible map f : T → T is called a permutation of T

• Symmetric Group: finite group of order n!:

Sn is the group of permutations of the indices 1, 2, . . . , n

• Subgroup: A subset H of group G such that:

a ∈ H, b ∈ H → ab ∈ H

1 ∈ Ha ∈ H → a−1 ∈ H

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• Circle group: set of complex number of absolute value 1 (C×)

• Special linear Group: subgroup of GLn(R)

SLn(R) : set of real n× n matrices A with determinant 1

• Every group has trivial subgroup (identity) and the subgroup G itself

1.3 Subgroups of the Additive Group of Integers

• Subgroup of Z+: (integers divisible by a)

Za = {n ∈ Z|n = ka for some k ∈ Z}

• Theorem: If S is a subgroup of Z+, then S is either the trivial subgroup {0} or has the form Za witha the smallest positive integer in S.

• Greatest Common Divisor: if ints a and b generate the subgroup S = Za + Zb, then S = Zd for thegreatest common divisor d of a and b.

• Relatively Prime: two nonzero ints a, b are relatively prime if their gcd: 1: iff there are integers r ands such that ra+ sb = 1.

• Least Common Multiple: integer m such that Za ∩ Zb = Zm

• Theorem: if d = gcd(a, b) and m = lcm(a, b), then ab = dm.

1.4 Cyclic Groups

• Cyclic Subgroup, H =< x > generated by x ∈ G:

H = {. . . , x−2, x−1, 1, x, x2, . . . }

• Theorem: S: set of integers k such that xk = 1 for < x >; then:

S ⊂ Z+

xr = xs iff xr−s = 1→ r − s ∈ S

if S 6= {1}, then S = Zn (n = | < x > |)

• Order of cyclic group is smallest positive integer n such that xn = 1

1.5 Homomorphisms

• Homomorphism ϕ : G→ G′: for all a, b ∈ G:

ϕ(ab) = ϕ(a)ϕ(b)

• Important examples:det : GLn(R)→ R×

σ : Sn → {±1}

ϕ(n) = an

• Trivial homomorphism: ϕ : G→ G′ that maps all elts of G to idG′

• Inclusion map i : H → G defines i(x) = x if H ⊂ G

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• ϕ(1G) = 1G′ , ϕ(a−1) = (ϕ(a))−1

• Image, Kernel (each subgroups of G′, G, kernel is a normal subgroup):

imϕ = {x ∈ G′|x = ϕ(a) for some a ∈ G}

kerϕ = {a ∈ G|ϕ(a) = 1}

• Important examples:ker(det) = SLn(R)

ker(σ) = An

• Left Coset: if H ⊂ G, a ∈ G:

aH = {g ∈ G|g = ah for some h ∈ H}

• Theorem: If ϕ : G→ G′ is a homomorphism, and a, b ∈ G, K = kerϕ, then the following are equivalent:

ϕ(a) = ϕ(b)

a−1b ∈ K

b ∈ aK

aK = bK

• A homomorphism is injective iff K = {1}.

• Normal Subgroup: a subgroup N of G is a normal subgroup, if for ∀a ∈ N , ∀g ∈ G, gag−1 ∈ N

• Center is always a normal subgroup of G:

Z = {z ∈ G|zx = xz ∀x ∈ G}

Example: center of SL2(R) is I,−I;Center of symmetric group of n ≥ 3 is trivial.

1.6 Isomorphisms

• Isomorphism ϕ : G→ G′ is a bijective group homomorphism

• If ϕ is an isomorphism, so is ϕ−1

• The groups isomorphic to a given group G form the isomorphism class of G

• Authomorphism: an isomorphism from a set to itself

• Examples of Aut:Conjugation by g:

ϕ(x) = gxg−1

• Conjugacy: gxg−1 is the conjugate of x by g; x and x′ are conjugate if x′ = gxg−1 for some g ∈ G

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1.7 Equivalence Relations and Partitions

• Partition: a partition of set S is a subdivision into disjoint, nonempty subsets

• Equivalence Relation:a ∼ b, b ∼ c→ a ∼ c (transitive)

a ∼ b→ b ∼ a (symmetric)

∀a, a ∼ a (reflexive)

• Theorem: An equivalence relation on set S determines a partition of S, and conversely

• Equivalence Class:Ca = {b ∈ S|a ∼ b}

• For any equivalence relation, there is a natural surjective map:

π : S → S

• Fibre: any map f : S → T gives equivalence relation: a ∼ b if f(a) = f(b); the fibres of the map f is:

f−1(t) = {s ∈ S|f(s) = t}

• Equivalence Relation defined by a Homomorphism: two elements are “congruent” iff their cosets areequal.

a ≡ b if ϕ(a) = ϕ(b)

• K = kerϕ, then fibre of ϕ that contains a ∈ G is aK, which partition G, corresponds to image of ϕ

1.8 Coset

• Left Coset of H:aH = {ah|h ∈ H}

• Cosets are equivalence classes for the congruence relation, and partition the group G

a ≡ b if b = ah for some h ∈ H

• Theorem: the three are equivalent:b = ah for some h ∈ H

b ∈ aH

aH = bH

• Index: number of left cosets of a subgroup:

[G : H]

• Theorem: All left cosets have the same order: because multiplication by a is a bijective map

• Counting Formula:|G| = |H|[G : H]

• Lagrange’s Theorem: if H ⊂ G, then |H| divides |G|

• Theorem: If |G| = p, a prime order, then if a ∈ G, G =< a > therefore forming just one isomorphismclass

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• Since left cosets of kernel are fibres, which have bijectvie correspondence with the image:

[G : kerϕ] = |imϕ|

• If ϕ : G→ G′ is a homomorphism of finite groups:

|G| = |kerϕ| · [G : kerϕ] = |kerϕ| · |imϕ|

|kerϕ| divides |G|

|imϕ| divides |G|, |G′|

• Right Coset: If a subgroup is normal, the left and right cosets are equal:

H is a normal subgroup

∀g ∈ G, gHg−1 = H

∀g ∈ G, gH = Hg

• Theorem: If H ⊂ G, g ∈ G, then gHg−1 ∈ G

• Theorem: If G has just one subgroup H of order r, that is normal

• If G is finite, indices of left/right cosets are the same.

1.9 Modular Arithmetic

• If n divides, b− a, (b = a+ nk)a ≡ b modulo n

• The congruence classes are the cosets of the subgroup Zn

a+H = {a+ kn|k ∈ Z}

• The set of congruence classes modulo n = Z/Zn

1.10 Correspondence Theorem

• Correspondence Theorem: Let ϕ : G → G be a surjectvie homomorphism with kernel K; there is abijective correspondence between subgroups of G and subgroups of G that contain K:

{subgroups of G that contain K} ↔ {subgroups of G}

1.11 Product Groups

• Product Set: set of pairs of elements (a, a′), with:

(a, a′) · (b, b′) = (ab, a′b′)

• Product Group: product of G and G′: G×G′

• If r and s are relatively prime, a cyclic group of order rs is isomorphic to the product of a cyclic groupof order r and a cyclic group of order s

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1.12 Quotient Groups

• Quotient Group: the set of cosets of a normal subgroup:

G/N is the set of cosets of N ∈ G

• Theorem: Let N be the normal subgroup of G, G the set of cosets of N , then there is a law ofcomposition on G that makes this set into a (quotient) group, such that π : G → G defined byπ(a) = a is a surjective homomorphism whose kernel is N

• Theorem: N is a normal subgroup of G; then, (aN)(bN) is also a coset: abN (Normality is crucial!)

• First Isomorphism Theorem: if ϕ : G → G′ is a surjective group homomorphism with kernel N , thequotient group G = G/N is isomorphic to the image G′. To be precise, let π : G→ G be the canonicalmap, then there is a unique isomorphism ϕ : G→ G′ such that ϕ = ϕπ

• Corollary: if ϕ : G→ G′ is a homomorphism with kernel N and image H ′, then the quotient group Gis isomorphic to H ′

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2 Vector Spaces

2.1 Fields

• Field: A field F is a set together with two laws of composition:

F × F → (+)F

F × F → (×)F

called addition and multiplication, such that:

1. Addition makes F into an abelian group F+, with identity = 0

2. Multiplication is commutative, so it makes the set of nonzero elements of F into abelian groupF× with identity = 1

3. Distributive law: ∀a, b, c ∈ F , a(b+ c) = ab+ ac

• Finite Field: ex. prime field: Z/pZ is a field

Fp = {0, 1, . . . , p− 1} = Z/pZ

• Cancellation Law: for a, b, c ∈ Fp,

1. ab = 0→ a = 0 or b = 0

2. a 6= 0 and ab = ac then b = c

• Theorem: if p is a prime integer, Fp is a field of order p (multiplicative inverse exists)Equivalently, if a is not divisible by p, there is an integer b such that ab ≡ 1 modulo p

• General Linear Group over Finite Fields:

GLn(Fp) = {n× n invertible matrices with entries in Fp}

SLn(Fp) = {n× n invertible matrices with entries in Fp with det = 1}

• Theorem: The characteristic of any field F is either zero or a prime number (characteristic: how manytimes you have to add 1 to get back to 0)

• Theorem: For p a prime integer, F×p is a cyclic group of order p− 1

2.2 Vector Spaces

• Vector Space: A vector space V over a field F is a set together with two laws of composition:

1. Addition: V × V → V : (v, w)→ v + w

2. Scalar Multiplication: F × V → V : (c, v)→ cv

with the following rules:

1. Addition makes V into an abelian group V + with identity 0

2. 1v = v ∀v ∈ V3. (ab)v = a(bv) ∀a, b ∈ F, ∀v ∈ V j4. (a+ b)v = av + bv, a(v + w), ∀a, b ∈ F , ∀v, w ∈ V

• Examples: if V = C, F = R; the set of real polynomials p(x) = anxn + · · ·+ a0; the set of continuous

real-valued functions on the real line; the set of solutions to the differential equation d2ydt2 = −y

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• Subspace: W ⊂ V is a subspace if it is a nonempty subset closed under addition and scalar multipli-cation, and includes a zero vector. It is a proper subspace if it is neither trivial nor V . (Kernel is asubspace!)

• If W is a proper subspace of R2, and w a nonzero vector in W , then W consists only of the scalarmultiples cw. Distinct proper subspaces have only the zero vector in common.

• Isomorphism from V to V ′

ϕ(v + w) = ϕ(v) + ϕ(w) and ϕ(cv) = cϕ(v)

2.3 Bases and Dimension

• Linear Combination: for S = (v1, . . . vn) (a hypervector), the linear combination of S is:

w = c1v1 + · · ·+ cnvn (ci ∈ F )

• Span: the set of all vectors that are linear combinations of S = (v1, . . . , vn) forms a subspace of V ,called the span of the set (smallest subspace that contains S)

• Column Space: the column space of an m×n matrix wwith entries in F is the supspace of Fm spannedby the columns of the matrix

• Theorem: If A is m × n matrix, B is a column vector, then AX = B has a solution iff B is in thecolumn space of A

• Linear Independence: an ordered set of vectors S = (v1, . . . , vn) is linearly independent if there is nolinear relation SX = 0 except for the trivial case X = 0

• Basis: A basis of a vector space is a set of vectors that is linearly independent and also spans V

• A vector space is finite dimensional if some finite set of vector spans it

• Theorem: If V is a finite-dimensional vector space, S is a subspace spanning V , L is an independentsubset of V , then we can obtain a basis by adding elements of S to L, or by deleting elements from S

• The empty set is independent; the span of the empty set is the zero space {0}

• Dimension: The dimension of a finite-dimensional vector space is the number of vectors in a basis

2.4 Computing With Bases

• Theorem: Every vector space V of dimension n over a field F is isomorphic to the space Fn of columnvectors

• Change of Basis: If B, B′ are the old basis, new basis, respectively, P is the basechange matrix:

B′ = BP

2.5 Direct Sums

• Sum: If W1, . . .Wk are subspaces of V , the set of vectors v that can be written as a sum: (wherewi ∈W )

v = w1 + . . . wk

Then the sum of the subspaces (or the span) is denoted as:

W1 + . . .Wk = {v ∈ V |v = w1 + · · ·+ wk, with wi ∈W}

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• Independence: the subspaces W1, . . .Wk are independent if no sum is zero, except the trivial sum:

w1 + . . . wk = 0 implies wi = 0 ∀i

• Direct Sum: If Wi sums to V and independent:

V = W1 ⊕ · · · ⊕Wk

If W1 + · · ·+Wk = V and W1, . . . ,Wk are independent

• If V is the direct sum, every vector in V can be written only in a unique way

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3 Linear Operators

3.1 Dimension Formula

• Linear Transformation: (analogous to a homomorphism) T : V →W is a map that is compatible with:

T (c1v1 + c2v2) = c1T (v1) + c2T (v2)

• Nullspace: The kernel of a linear transformation

• Dimension Formula: (rank-nullity Theorem) for T : V →W

dim(kerT ) + dim(imT ) = dimV

• The image of T is the column space if T is left multiplication by a matrix

• If determinant is nonzero, the nullspace is {0} and rank = n; if determinant is zero, rank < n

3.2 Matrix of Linear Transformation

• Theorem: T : Fn → Fm is a linear transformation between spaces of column vectors, and let thecoordinate vector of T (ej) be Aj = (a1j , . . . , amj)

t. Let A be the m × n matrix whose columns areA1, . . . , An. Then T acts on vectors in Fn as multiplication by A.

3.3 Linear Operators

• Linear Operators: Linear transformations T : V → V that map a vector space to itself

• Example: Left multiplication by a square n× n matrix with entries in F defines a linear operator onthe space Fn of column vectors

• Rotation Matrix: Defines a counterclockwise rotation of the plane through angle θ

R =

[cosθ −sinθsinθ cosθ

]• Theorem: Let K, W denote the kernel and image of a linear operator T on V :

– The following conditions are equivalent:

1. T is bijective

2. K = {0}3. W = V

– The following conditions are equivalent:

1. V = K ⊕W2. K ∩W = {0}3. K +W = V

• Chenge of Basis: Let A be the matrix of linear operator T wrt basis B; let B′ be a new basis such thatB′ = BP , then:

A′ = P−1AP

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3.4 Eigenvectors

• T-Invariance: A subspace W of V is T − invariant if it is carried to itself by the operator:

TW ⊂W

• Eigenvector: an eigenvector v of a linear operator T is a nonzero vector:

T (v) = λv

where λ is the eigenvalue

• If v is an eigenvector of T , with eigenvalue λ, the subspace W spanned by v will be T -invariant, becauseT (cv) = cλv

• Eigenvector is a basis of a one-dimensional invariant subspace

• The matrix of T wrt a basis B = {v1, . . . , vn} is diagnoal iff each basis vector vj is an eigenvector

• Positive matric: Matrix whose entries are all positive; they always have a positive eigenvector

3.5 Characteristic Polynomial

• Singular Operator: a lienar operator that is not invertible

• The following are equivalent:

1. T is a singular operator

2. T has a zero eigenvalue

3. If A is the matrix of T , then detA = 0

• Characteristic Polynomial: of a linear operator T is the polynomial

p(t) = det(tI −A)

• The eigenvalues of a linear operator are the roots of its characteristic polynomial

• Theorem: the characteristic polynomial of an n× n matrix A has the form:

p(t) = tn − (trace A)tn−1 + · · ·+ (−1)n(detA)

• The characteristic polynomial, the trace, and the determinant are independent of the basis

• Theorem: If λ1, . . . λn are the eigenvalues of n× n complex matrix A, then:

detA = λ1 . . . λn

trace(A) = λ1 + · · ·+ λn

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4 Applications of Linear Operators

4.1 Orthogonal Matrices and Rotations

• Dot Product: If X = (x1, . . . , xn)t, Y = (y1, . . . yn)t, then

(X · Y ) = x1y1 + . . . xnyn

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5 Lecture Notes Confusing Concepts

5.1 Matrices

• Pivot: first non-zero element in each row

• Row-Echelon Form:

1. Pivot has entry 1

2. Pivot of each row is right of previous row

3. Pivot has 0s above

Example:

1 0 30 1 20 0 0

• Elementary Row Operations (Ei)

1. Row Addition

2. Row Swap

3. Row Scale

• If M ′ is the REF of M , then M ′ = I or M ′ has bottom row of all 0s

5.2 Groups

• Sn is non-Abelian for n ≥ 3

• On(R) ⊂ GLn(R) ⊂ Aut(Rn)

• Euclidean Theorem: The only subgroups H ⊂ Z are those of the form nZ

• Group Action: G acts on a set S, if ∀g ∈ G, we have a bijection f(g) : S → S and f(gh) = f(g)f(h)Alternate definition: a map G × X → X: (g, x) → π(g)x such that π(e)x = x and π(g1g2)x =π(g1)[π(g2)x]

• Example: if S = {1, . . . , n}, then Aut(S) = Sn, and f : G→ Sn is called the permutation representa-tion of GGLn(R) acts on the set S = Rn, G acts on itself by left translation or conjugation

• Orbit: orbit of s is everything that can be reached from s by an action of something in G

Orbit of s = {s′ = g(s) for some g ∈ G}S = ∪ Orbit s (disjoint union)

• Center: If f : G→ Aut(G) such that g → h→ ghg−1, then

ker(f) = Z(G) = {g ∈ G : gh = hg ∀ g ∈ G}

• Stabilizer:Gx = {g ∈ G|g(x) = x}

• For the group action by conjugation, f(g) : G→ G is a group isomorphism

• If H ⊂ G is stable under conjugation, it is a normal subgroup of G

• Simple: If the only normal subgroups of G are G and {e}, we say G is simple

• An is simple for n ≥ 5

• H is a normal subgroup of G iff H is a union of conjugacy classes

• The kernel of a homomorphism f : G→ G′ is a normal subgroup of G; in fact, every normal subgroupis the kernel of some group homomorphism

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5.3 Equivalence Relation

• Examples of Equivalence Relation:

1. Orbits of a group action

2. Fibres of a map of sets (S → S′): if image = T ⊂ S′ then fibre(t) = St

G =⋃Gt

3. Homomorphism of groups (f(s) = f(s′))

• For f : G→ G′, fibres give equivalence relation, s ∼ s′ iff s′ ∈ sHOr equivalently, s ∼ s′ iff s−1s′ ∈ H and the cosets sH are all isomorphic as sets to H

• Lagrange’s Theorem: |G| = |H|[G : H]

• If |G| is prime, then the only subgroups of G are {e} and G

5.4 Cosets

• If H is a normal subgroup of G, gH = Hg

• Group Law on G/H:aH · bH = (ab)H

• Then, the surjective homomorphism f : G→ G/H with ker(f) = H is defined

• If G is abelian, then every H ⊂ G is normal, and G/H is an abelian group

• Theorem: Any cyclic group G is isomorphic to either (Z+) if it is infinte, or (Z/Zn, (+)) if it is finiteof order n

• The only simple abelian groups are Z/Zp for prime p

• (Burnside) For every nonabelian finite simple group G, |G| is divisible by at least 3 primes

• (Feit-Thompson) For every nonabelian finite simple group G, |G| is even

• Example: GL3(Z/Z2) is simple of order 168 = 23 · 3 · 7

• Define: the classes with multiplicative inverses ( modulo n):

(Z/Zn)∗ = {classes relatively prime to n}

• Euler’s Theorem: if p is prime, (Z/pZ)∗ = Z/pZ− {0} is the cyclic group of order p− 1

5.5 Fields

• For any field, F , there is a group homomorphism f :

Z → F

0→ 0

1→ 1

n = 1 + · · ·+ 1→ 1F + · · ·+ 1F

• ker(f) = dZ, such that either d = 0 and f is an injection, or d is a prime number

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• S ⊂ V is a basis for V if it satisfies one of :

1. S spans V , but no smaller subset of V spans V

2. S is linearly independent, but no larger subset of V is linearly independent

3. S spans V and is linearly independent

• Quotient Space: V/W :

1. Quotient group on cosets v +W of W in V :

(v +W ) + (v′ +W ) = (v + v′) +W

2. Scalar multiplication on cosets:c(v +W ) = cv +W

• dim(V ) = dim(W ) + dim(V/W )

• T : V →W is a linear isomorphism and {v1, . . . , vn} is a basis of V iff {Tv1, . . . T vn} is a basis of W

• Theorem: A linear map T : V → W is completely determined by the vector Tv1, . . . , T vn in W for{v1, . . . , vn} a basis of V

• If dimV = n, there is a linear isomorphism V → Fn

• Hom(V,W ) = {all linear maps T : V →W} is isomorphic to Fmn

• T + S(v) (in )Hom(V,W ) = Tv + Sv (in )W

• If V = W , we call Hom(V, V ) = End(V ), and Aut(V ) ⊂ End(V )

• V 6= W ⊕ V/W as V/W is not a subspace

• There may be cases where there are no eigenvectors at all

5.6 Change of Basis

• B = P−1AP

• det(A) = det(B) even if basis changed

• T : V → V is a linear isomorphism iff det(T ) 6= 0 in F

• G = GL(V ) is a group of all invertible T : V → V under composition

• For det : GL(V )→ F ∗, this is a surjective group homomorphism

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