Review Sheet Math 581

[n] means the topic is found on acetate n.   Acetates after  [80]  are new to the review sheet after the midterm.

Definitions

Group [2]

Identity element

Inverse element [2]

Abelian group [20]

nonAbelian group [20]

Examples:

Groups of permutations [3]

Zn under + [25]

Cyclic groups [29 to 33.2]

Groups of small order, like 1,
2, 3, 4, 5 [39, 40, 47, 48, 49]
 

Generators and relations for a group Symmetry groups of n-gons [83.0-84, 85, 86]
Cosets, left and right [41] Examples [41, 42, 43]

Subgroup (g) generated by an element g  [29]

Cyclic group [32]

Order of a group [38]

Order of a group element [38]

Examples in a group of symmetries of a geometric figure of the left and right cosets of the subgroup fixing a point of the figure. [83.0-84, 87]

Index (G: H) of a subgroup H in a group G. [46.1]

Function

Well-defined

Onto

One-to-one

Permutation of a set [3]

n-cycle [8]

transposition (2-cycle) [7]

even permutation [8.9]

odd permutation [8.9]

Isomorphism of groups [24]

Subgroup [27.0, 27.1]

Homomorphism of groups [57]

Examples [58 to 60.2]

Kernel of a homomorphism [61.3]

Examples using symmetry groups  [88-94]

Normal subgroup [66]

Quotient group G/H of a group G over a normal subgroup H. [67 to 70]

Symmetry of a geometric figure [75]

Examples [76, 78 81-84]

Symmetry group of a cube [98.9-99.93]
 

Vector space over a field  F [101-102]
Vector, scalar, zero vector, vector addition, scalar multiplication  [101-103]

Scalar multiple  [109]

Linear combination [110, 116]

Spanning set [117]

Examples  R[100],   Fn   [112]

Polynomials of degree   £ m. [111]

The set of all polynomials  F[x]  with coefficients in a field  F. [112]

Isomorphic vector spaces [154]

Subspace  [107]
Examples [109-110]

Subspace spanned by a set of vectors [110]

The span  (t1, ...,tm)  of a set of vectors  {t1, ...,tm}.  [110.1]

The span  (T) of a set of vectors  T  [112]

The intersection and the sum of two subspaces  [114]

Linearly independent set of vectors  [118, 125]

Linearly dependent set of vectors [119, 121.1]

Examples of linearly independent and linearly dependent sets of vectors [119-120]

Basis [126]

Finite-dimensional vector space [130]

Dimension of a finite-dimensional vector space [132.0]

Linear transformation  [141]

Examples [141-145]

Equal linear transformations [147]

Kernel of a linear transformation [150]

Image of a linear transformation [150]

Isomorphism of vector spaces [154]

Scalar multiplication of linear transformations  [162-163]

Composition of linear transformations [166-170]

Matrices
Matrix of a linear transformation  with respect to two given bases [156]

Matrix [157]

Addition of matrices  [159-161]

Scalar multiplication of a matrix  [162-163]

Multiplication of matrices [166-170]

 
Theorems--with proofs The set of all permutations of a set S with multiplication equal to composition of functions is a group. [4.0 to 4.5]

Every permutation is a product of transpositions [7 to 8.0]

The identity element in a group is unique [20]

The inverse of a given group element is unique [21]

(a')' = a [21]

(ab)' = b'a' [22]

If ab = ac, then b = c [22]

If ba = ca, then b = c [22]

If a is a fixed element of a group G, then the function F which maps g in G to ga, is a permutation of G. [ 23]

If H is a subgroup of G, then the identity element of g is in H. [27.1]

If H is a nonempty subset of a group G with the property that for every h, k in H, hk' is in H, then H is a subgroup of G. [28]

The set of all powers of a fixed element  g  is a subgroup. [29]

A cyclic group is Abelian [30]

If gk = gm for different exponents k and m, then  (g)  is a finite cyclic group and the number of elements in  (g)  is equal to the least positive integer  n  for which  gn = e. [31]

If H is a subgroup of G, then

If a is in H, then aH = Ha = H;

If a and b are in G, then aH and bH are either disjoint or equal (as subsets of G) (Same for Ha and Hb.)

All cosets, left and right have the same number of elements. [44 to 46.0]
 

LaGrange's Theorem [46.1]

If F is a homomorphism from G into C, then

F maps the identity of G to the identity of C [61.0],

F(a') = F(a)' for every a in G [61.0]

The kernel of a homomorphism is a normal subgroup. [61.4]

A homomorphism is one-to-one if and only if its kernel is {e}. [65]

If H is a normal subgroup of a group G, the mapping F from G onto G/H given by F(a) = aH is a homomorphism with kernel H. [71.0, 71.1]

Let  j  be a homomorphism from a group  G  into a group  C.  Let  K  be the kernel of  and  j(G be the image of  G.  Let    be the natural homomorphism of  G  onto  the quotient group  G/K. Then there is a unique isomorphism  F  of   G/K  onto  j(G)  such that  aF  =  j.  [95-98]
(On the acetates  F  is denoted by  j  wearing a hat.)

Elementary properties of vector spaces  [104-106]

The span of a set of vectors is a subspace [110-112]

The intersection and the sum of two subspaces are subspaces [114]

If  {v1,  ... , vm}  is a linearly independent set of vectors, then every vector in its span has a unique expression as a linear combination of  v1,  ... , vm.  [121.0]

If  T  is a finite set of vectors written in any order, then  T is linearly dependent if and only some vector in  T  is a linear combination of the vectors in  T  which precede it. [121.2-121.3]

{e1,  ... , en}  is a basis for  Fn. [122-123]

{1, x, ... , xm - 1}  is a basis for the vector space of all polynomials of degree  £ m - 1. [124]

If  m > 0  and  {v1,  ... , vm}  is any  set of vectors  in a vector space  V, and if  {w1,  ... , wn}  are  n  vectors in  v  where  n  > m, then   {w1,  ... , wn}  is a linearly dependent set of vectors. [127-129]

If  V  is a finite-dimensional vector space, then all bases of  V  are finite and have the same number of elements. [130-131]

If  W and  W2 are subspaces of a finite-dimensional vector space  V, then  dim (W1) + dim (W2) =
dim (W + W2) + dim (intersection of  W and  W2). [137-140]

Elementary properties of linear transformation [146]

If  V  is a vector space with basis  {v1,  ... , vn}  and  {w1,  ... , wn} is any  set of vectors  in a vector space  W,  then there is a unique linear transformation  A:  V  Æ  W  such that   "i   viA =  wi.  [148-149]

The kernel and the image of a linear transformation are subspaces [150]

If  A  is a linear transformation of a finite-dimensional vector space  V  into a vector space  W, then
dim (V)  =  dim (kernel A) + dim (image A). [151-152]

A linear transformation is  one-to-one  if and only if its kernel is the zero vector alone. [153]

Two finite-dimensional vector spaces over  F are isomorphic if and only if  dim (V) = dim (W). [154-155]

The sum of linear transformations,  scalar multiples of linear transformations, and composition of linear transformations are again linear transformations [159-170]
 

Theorems--no proofs Every permutation of a finite set can be written as a product of disjoint cycles [5.5]

Every permutation is either even or odd, but not both [8.9 to 14]

Every cyclic group is either infinite and isomorphic to Z, or is finite and, if it has n elements, is isomorphic to Zn. [30 to 33.0]

Every subgroup of a cyclic group is cyclic [34 to 35]

Know what are all the subgroups of a cyclic group [36 to 37.1]

Every group of prime order p is isomorphic to Zp, + [49]

Every group of order 4 isomorphic to either Z4 or Z2XZ2. [47, 48]

Every group of order 6 isomorphic to either Z6 or S3. [50 to 56.3]

If H is a subgroup of G, then aH = bH if and only if a'b is in H. [56.4]

If F is a homomorphism from G into C, then

F (ak) = F(a)k for every a in G and every integer k [61.0];

if H is a subgroup of G, then F(H) is a subgroup of C. [61.1]

If H is a subgroup of G and g is a fixed element of G, then g'Hg is a subgroup of G and is isomorphic to H. [61.2]

A normal subgroup is the kernel of some homomorphism. [66]

A symmetry of a polygon or polyhedron is completely determined by how it permutes the vertices. [77]

The set of symmetries of a geometric figure is a group with multiplication equal to composition of functions. [79, 80]

A subspace of a vector space is a vector space [108]

In a finite-dimensional vector space  V, every linearly independent subset can be extended to a basis for  V.[132.0-132.1, 133, 134, 135.0]

If  W  is a subspace of a finite-dimensional vector space  V, then  W  is finite-dimensional
and  dim(W)  £  dim(V). [135.2]

Each of  L(V; W)  and  F(n, m)  has dimension equal to
dim (V) ¥ dim (W)  [164]
 
 

Calculations Write any permutation as a product of disjoint cycles [5.5]

Calculate products using cycle notation [6]

Write any permutation as a product of transpositions [7 to 8.0]

Determine whether a permutation is even or odd [8.9]

Determine whether or not a given configuration of the fifteen puzzle can be reached from the initial configuration. [16 to 19]

Calculations with generators and relations of a group. [83-86, 98.9-99.93]

Extending a linearly independent set to a basis of a vector space [135.1]
 
 

Notation e for identity element [2]

a' or a-1 for the inverse of a [2]

composition of functions, read left to right [3]

aF for F(a) [4.1, 4.2]

Two-row notation for permutations [5]

Cycle notation for permutations [5.0]

Notation for isomorphism [25]

(g) for the cyclic subgroup generated by  g. [29]

R for the vector space of n-tuples of real numbers. [100]

F for the vector space of n-tuples of elements in a field  F. [122]

(t1, ...,tm)  for the subspace spanned by the set of vectors  {t1, ...,tm}.

dim (V)  for the dimension of a vector space. [132.0]

notations for matrices [157]

L(V; W)  and  F(n, m)  [164]

The matrices  Eij  form a basis for  F(n, m)  [165]