vector space

a vector space is a set of vectors such that two vectors can be added to form another vector in , and a vector can be multiplied by a scalar to form another vector in . this addition and multiplication must satisfy these rules:

1. additive identity. there exists a vector such that for any , we have .
2. additive inverse. for any , there exists a vector such that .
3. commutative law for addition. for all , we have 4. associative law for addition. For all , 5. multiplicative identity. for all , we have .
4. associative law for multiplication. for all scalars and all , 7. distributive law for scalar addition. for all scalars and all , we have .
5. distributive law for vector addition. for all scalars and all , we have .

[cite:;taken from @calc_hubbard_2015 definition 2.6.1]

a broken link: blk:def-vector-space over is called a real vector space.
[cite:;taken from @algebra_axler_2024 chapter 1 vector spaces; definition 1.22]

a broken link: blk:def-vector-space over is called a complex vector space.
[cite:;taken from @algebra_axler_2024 chapter 1 vector spaces; definition 1.22]

a vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors. the unique integer such that every basis for contains exactly elements is called the dimension of and is denoted by a vector space that is not finite-dimensional is called infinite-dimensional.
[cite:;taken from @algebra_insel_2019 chapter 1 vector spaces]

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 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