Vector space is a word that where in two different fields of study Physics and Mathematics. So What is a vector space and axioms? which component is called vector space? Basic definition of vector space is a set of elements, named vectors which can be added to each other vectors and can be multiplied by scalar.

Scalar is usually a real number, sometimes the scalar value can be complex numbers. To be a vector space element vector addition and scalar multiplication should satisfy a few necessities called an axiom.

Depending on the nature of the scalar the vector space is differentiated as real vector space if the scalar is real coordinate space and the vector space is complex vector space if the scalar is complex coordinate space. Now everyone has a basic question regarding vector space is that why do you use the word vector for a set of an element that satisfies axiom?  

Vector is a Latin word that means carries. The answer to why the word vector is used for a set of elements has a history behind it, which started in from 18th century by astronomers who defined the word vector for the motion of the planets. According to them, a vector means an object that carries from one point to another. It is multidimensional which has length and direction whereas scalar has only length. In the 19th century vector space was used for a set of elements where certain operations can be defined. Matrix though is different from a common vector in physical space, the matrix can be added and multiplied so they form vector space.

Euclidian vectors are physical quantity that has both direction and magnitude. Example of Euclidian vectors is force and velocity. The generalization of Euclidian vectors is called vector space. This vector space which is fundamental for linear algebra together with matrix provides an efficient way of manipulation in the system of linear equations. 

Vector space is categorized by the dimension which is the number of independent directions in the space. Technically vector space is isomorphic which means they have an equal structure in the space. Depending on the dimensions, vector space is either a finite dimension or an infinite dimension. A vector space is said finite dimension if the dimension is the natural number, vector space in geometry is probably finite and the vector space is an infinite dimension if the dimension is the infinite cardinal, the vector space in the area of mathematics is like a polynomial ring is probably infinite. 

Vector Addition

Before going deep into vector space let me give you an example of vector addition for a clear understanding.

Initially, the vector is the combination of magnitude which is the value of the vector and direction so this vector cannot be specified as the normal value. The vector is specially denoted by an arrow above the variable that denotes the vector.

Let x and y be two vectors and the addition of two vectors is just x +y.

Vector x =xxi + xyj + xzk and y = yxi + yyj + yzk.

Then the resultant vector is x + y = (Xx + Yx)i + (Xy + Yy)j + (Xz +Y z) k few conditions need to be satisfied for vector addition

• The same vector can only be added for example an acceleration will only be added to an acceleration, not with any mass value.
• A vector cannot be added to a scalar value. 

Vector Multiplication

Vector Multiplication is the product of two vectors or the multiplication of vector and scalar value. Product of two vectors of two types

• Dot product of two vectors is also called the scalar product as the resultant of two vector dot product is the scalar value.
• Cross product of two vectors is also called vector product as the resultant of two vector cross product is also a vector. The resultant vector is perpendicular to the other two vectors. 

Let a be a vector and b be another vector, x be a scalar value multiplying the scalar value to the b vector, and adding these two vectors is a+xb. 

Is vector and Vector space same?

Vector is a part of vector space and vector space is a group of elements called vectors and manipulated by the scalar. Ultimately the combination of the vector is vector space.  

Equal Vectors

Two vectors are said to be equal if their direction and magnitude are the same which means if it is drawn as a line segment they will be parallel to each other. 

Axioms of Vector Space

Axioms are the set of rules or principles that most people believe to be true. According to maths, an axiom is a set of rules or principles which is unprovable but accepted as true rules as it is self-evident. Vector addition and vector Multiplication need to satisfy a few axioms

Here is the list of axiom that vector addition and vector Multiplication need to satisfy. 

Vector Addition axioms

Let ‘a’ and ‘b’ are the two vectors in vector space, and let the resultant be ‘c’. The addition of two vectors will be “c=a+b” here ‘a’ and ‘b’ vector is said to belong to vector space ‘c’.

Axiom of vector Addition are

• Commutative Law: For all vectors ‘a’ and ‘b’ in vector space ‘c’, then it should follow the statement a+b =b+a.
• Associative Law: For all vectors ‘a’ and ‘b’ in vector space ‘c’, then it should follow the statement a+(b+c)=(a+b)+c.
• Closure: If any vectors are present in the vector space then the addition of these vectors also belongs to the vector space. So let ‘a’ and ‘b’ are the vectors that belong to the ‘c’ vector space then a+b the resultant belongs to vector space ‘c’.
• Additive Identity: Additive Identity is usually ‘0’. For a vector ‘a’ in the vector space ‘c’ then the statement that vector addition needs to satisfy is a+0=a and 0+a=a.
• Additive Inverse: For a vector ‘a’ in the vector space ‘c’ then there exists an additive inverse of vector ‘a’ is -a.

Vector Multiplication Axioms

Here the vector Multiplication is done between the vector in the vector space and a scalar value or real numbers. The multiplication operation should satisfy the following rules. Get related articles here.

Axiom of Vector Multiplication

• Associative Law: For a vector ‘x’ in the vector space C and the real numbers be ‘u’ and ‘v’, then the statement is u.(v.x)=(u.v).x.
•  Distributive Law with respect to vectors: For a vector ‘x’ and ‘y’ in the vector space C and the real number be ‘u’, then the statement is u.(x+y)=u.x+u.y.
• Distributive Law with respect to vectors: For a vector ‘x’ the vector space C and the real numbers be ‘u’ and ‘v’, then the statement is x.(u+v)=u.x+x.v.
• Closure: For a vector ‘x’ in the vector space ‘C’ and the real number be ‘u’ then x.u belongs to the vector space ‘C’.
• Unitary Law: It is similar to the Additive Identity of vector addition. Here ‘x’ is the vector in the vector space, and the Unitary denotes 1, then the statement is x.1=x and 1.x=x. 

Properties of Vector Space

Properties of vector space are extracted from the axiom that it follows. Here is the list of properties of vector space.

• Negative of 0 is 0 which is -0=0.
• If the addition of two vectors is zero then one vector is the negative value of another vector that is if a+b=0 then b is equal to -a.
• Addition operation of a series of vectors that is finite can be of any order. The resultant will not change if the order of vectors, in addition, an operation is changed.
• Let a be a vector of vector space C then the negation of a which -a will also belong to vector space C.
• Vector Multiplication of any vector in the vector space with zero will give zero as the resultant. That is x is a vector then the product of x and zero is zero x.0=0.
• Vector Multiplication of real number and the zero vector is zero. That is let c be the real number then c.0=0.
• If the product of a scalar value and the vector in the vector space is zero. Then either the scalar or the vector will be zero. That is let c be the scalar value and x be the vector in the vector space and if c.x =0 then either c or x will be zero.
• Let x and y be the two vectors of the vector space. If x+y=x then obviously y will be zero. Therefore 0 is the only vector that is 0.
• Vector Multiplication of the value -1 with any vector in the vector space is the negation of the vector. That is -1. (x)=-x where x is the vector of the vector space. 

Subspace of the Vector Space

Subspace is a vector space that is contained in a vector space. Ultimately subspace is also a vector space but it is defined in comparison to the other vector space. Some more details from here.

Definition: Let x be the vector space and y be the subset of the vector space of x. Then y is the subspace of x under certain conditions. In simple terms, the subspace is the vector space with a smaller dimension.

Let x be the vector space that is the subset of y which is the vector space then to find if a vector space is the subspace of another vector space it should follow certain conditions.

• closure under addition : x , y€ Y the x+y€Y
• closure under multiplication: a is the real number then a.x€Y
• additive Identity:0€Y.