a vector space is a set
along with an addition on
and a scalar multiplication on
such that the following properties hold.
- commutativity
. - associativity
and for all and for all . - additive identity
there exists an element such that for all . - additive inverse
for every , there exists such that . - multiplicative identity
for all . - distributive properties
and for all and all .
a vector space over
is called a real vector space.
a vector space over
is called a complex vector space.
some stuff from college
let
be a non-empty set of vectors containing numbers from the field
, consider the 2 operations addition and scalar multiplication,
is a vector space only if it abides by the following axioms:
addition axioms:
- addition closure: for every
we have
- associative addition: for every
we have
- commutative addition:
- zero vector:
so that
- negative vector: for every
there exists
so that
multiplication axioms:
- multiplication closure: for every
and
we have
- associative multiplication: for every
and
we have
- identity vector: for every
we have
- identity law: for every
we have
- first distributive law: for every
and
we can have
- second distributive law: for every
and
we can have
over some field
over some
and some field
:
the definition of summation would be:
and for some
the definition of multiplication would be:
this is an example of the 1st addition axiom
let
be a field
this describes all the polynomials of
over the field
let
as an example
the addition of 2 polynomials is as follows:
the symbolic process of addition can be described as follows:
let
so there exist the polynomial degrees
and the symbolic process of constant multiplication is defined as:
as for the degrees of the resulting polynomials after multiplication/addition:
if
:
if
:
a vector space could look something like this:
all the vectors that lie on the blue line represent a vector space, because the multiplication of a line would just make it longer (or shorter) it wouldnt make it move out of the blue line, and addition of any 2 vectors that lie on the blue line would also result in a longer (or shorter) vector that lies on the same line which expands across the entire 2d space
this line doesnt represent a vector space because it doesnt contain the vector
reduction law
for every
for
:
for
for
and