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