ABSTRACT MULTILINEAR
ALGEBRA
BY
GYULA RÁBAI
WHAT IS A NUMBER?
WHAT IS A NUMBER?
An element of :
ALGEBRAIC STRUCTURE OF NUMBERS
Commutative Ring
Field
Free Commutative Monoid
SET
There exists a set M [Name]
NON-EMPTY SET = SET + ELEMENTS
There exists a set M
where a and b are elements of M,
[Name]
[Elements]
PARTIAL MAGMA = NON-EMPTY SET +
BINARY OPERATION
There exists a set M
where a and b are elements of M,
[Name]
[Elements]
and there exists an operation on some a and b
[Bin-arity]
MAGMA = PARTIAL MAGMA + TOTALITY +
CLOSURE
There exists a set M
where a and b and c are elements of M,
[Name]
[Elements]
and there exists an operation on every a and b
that results in c
[Total Bin-arity]
[Closure]
MAGMA = PARTIAL MAGMA + TOTALITY +
CLOSURE
There exists a set M with elements a and b [Set]
The result of an operation on any a and b is in M
[Total Binary Closure]
UNITAL MAGMA = MAGMA + IDENTITY
There exists a set M with elements a and b [Set]
The result of an operation on any a and b is in M
[Total Binary Closure]
There exists an identity element e such that e
compose any other element is itself. [Identity]
MONOID = UNITAL MAGMA + ASSOCIATIVITY
[Set]
[Total Binary Closure]
[Identity]
[Associativity]
MONOID EXAMPLE -
COMMUTATIVE MONOID = MONOID +
COMMUTATIVITY [Set]
[Total Binary Closure]
[Identity]
[Associativity]
[Commutativity]
COMMUTATIVE MONOID –EXAMPLE N
MOD 4
FREE MONOID OF A COMMUTATIVE
MONOID
NATURAL NUMBERS = FREE COMMUTATIVE
MONOID
Counting numbers without counting!!!
WHERE ARE WE?
COMMUTATIVE MONOID
COMMUTATIVE GROUP = COMMUTATIVE
MONOID + INVERSIBILITY
[Inversibility]
COMMUTATIVE GROUPS
Monoid:
Group:
WHERE ARE WE?
FIELD = C GROUP + C GROUP +
DISTRIBUTIVITY
[Distributivity]
WHERE ARE WE?
RING
USEFUL AXIOMS
Associativity
Commutativity
Identity
Distributivity
(Inversibility)
WHAT IS A VECTOR?
VECTOR SPACE = ACID + (BASE) FIELD
Associativity
Commutativity
Identity
Distributivity
Also inverse with scaling by -1; Associative with scalar mul; neutral to 1x but… could not be bothered to latex it)
MATRICES ARE VECTORS! (RUNNING OUT
OF LATEX)
M+0 = M
M+N = N+M
(M+N)+P= M+(N+P)
s(M+N)= sM + sN
(real proof where we define a matrix as a mapping between V is
pretty much the same)
Polynomials, waves, continuous functions, arrows, forces and many more are also vectors
LINEAR ALGEBRA IN 2 SLIDES ^^^
MULTILINEAR ALGEBRA?
•Tensor product (x)
•Distributivity ((u+v) prod w = …) and (u prod (v+w) = …)
•Scalars move anywhere.
DONE
