Overview 18.701
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Fall 2004
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Material in this course: Everything discussed in the lectures which
corresponds roughly to the following sections of the book
Chapter | Sections
--------------------------------
I | all
II | all
III | all but 5
IV | all but 5,7,8
V | 5,6,7,8
VI | 1,2,3,4,6
X | all but 8
XI | 1,2,3,4
XII | all but 8
Things you should know how to define (at least,there may be more this
seems like a fairly complete list which I got by simply reading
through my notes, in other words I did your preparation for you lewt
me know if I should add something)
matrix
vector
operations on matrices and vectors
inverse matrix
elementary row operations
elementary matrices
system of linear equations
determinant function, determinant
adjoint matrix
group
permutation group
symmetric group
cycle (notation)
permutation matrices
sign of a permuation
subgroups
subgroup generated by
cyclic (sub) group
order of element
order of group
Klein four group
Quaternion group
homomorphism of groups
isomorphism of groups
isomorphic groups
automorphisms of groups
inner automorphisms & conjugation
kernel & image of a homomorphism of groups
conjugates
normal subgroups
central element of a group
center of a group
relation on a set
equivalence relation
partition on a set
equivalence class
equivalence relation defined by a map
cosets (left and right)
congruence relation
products of groups
quotient groups
field
prime field
characteristic of a field
vector space
subspace
linear map or linear transformation
endomorphism
isomorphism
automorphism
kernel
image
subspace spanned by
linear combination
V spanned by S
linearly (in)dependent
basis
coordinates
finite dimension(al)
direct sums
matrix of T with respect to bases
matrix of change of base
determinant of a linear endomorphism
eigenvector
eigenvalue
invariant subspace
diagonal matrix
similar matrices
diagonalizable matrix (or linear endomorphism)
characteristic polynomial of a linear endomorphism
scalar matrix
more group
operation or action of a group on a set
orbit
transitive action
stabilizer
fixed point
centralizer of x \in G
simple group
solvable group
Sylow p-subgroup or p-Sylow subgroup or ...
ring
commutative ring (OK: all rings are commutative)
subring
homomorphism of rings
isomorphism
automorphism
kernel
image
ideal (of a ring)
principal ideal
ideal generated by
quotient ring
maximal ideal
special elements
unit
nilpotent
zero divisor
prime element
irreducible element
Gaussian integers
polynomials
polynomial over a field
polynomials over rings
polynomial rings
degree of a polynomial
evaluation of a polynomial
constant polynomial
monomial
multi-index
ring (continued)
integral domain
fraction field
maximal ideal
factorization (happens in domains)
prime element
irreducible elements
factorization
Noetherian ring
Principal ideal domain
Euclidean domain
Unique factorization domain
module
free module
submodule
homomorphism
isomorphism
endomorphism
automorphism
kernel
image
submodule generated by
elements generating M
fintely generated
presentation of M
classification of finite dimensional vector spaces endowed with an
endomorphism: rational canonical form, Jordan block, Jordan canonical
form