Table of contents  
Introduction  
Definition  
Vector Space Axioms  
Vector Space Properties  
Vector Space Problems 
A vector space, also known as a linear space, is a collection of objects called vectors. Vectors can be combined through addition and scaled by numbers, typically real numbers, known as scalars. However, in some instances, scalar multiplication can involve rational numbers, complex numbers, and other numeric types. The operations of vector addition and scalar multiplication must adhere to specific mathematical rules, often referred to as axioms. Real vector spaces and complex vector spaces are terms used to denote the use of real or complex numbers as scalars in these spaces.
A vector space, characterized by the presence of vectors, adheres to specific mathematical properties related to vector addition and scalar multiplication. These properties include the associative and commutative laws for vector addition and the associative and distributive rules for scalar multiplication.
In a vector space, you'll find the following components:
Both vector addition and scalar multiplication operations must meet specific conditions. These conditions are outlined in axioms, which extend the properties of vectors found within the field F. When a vector space is defined over the real numbers (R), it is termed a real vector space, while over the complex numbers (C), it is known as a complex vector space.
What distinguishes a Vector from a Vector Space?
A vector is a constituent element of a vector space. On the other hand, a vector space is a collection of objects that can be scaled by scalars and subjected to vector space axioms.
Is zero considered a vector space?
Indeed, zero qualifies as a vector space, known as the trivial vector space, denoted by {0}. This space contains the zero vector or null vector, and in this scenario, vector addition and scalar multiplication are straightforward and minimal.
Defining Equal Vectors
Equal vectors are vectors that share both the same magnitude and the same direction. When two vectors are considered equal, it means the line segments they represent are parallel, and their vector columns exhibit identical characteristics.
All the axioms should be universally quantified. For vector addition and scalar multiplication, it should obey some of the axioms. Here eight axiom rules are given.
An operation vector addition ‘ + ‘ must satisfy the following conditions:
Closure : If x and y are any vectors in the vector space V, then x + y belongs to V
An operation scalar multiplication is defined between a scalar and a vector and it should satisfy the following condition :
Closure: If x is any vector and c is any real number in the vector space V, then x. c belongs to V
Here are some basic properties that are derived from the axioms are
x − y = x + (−y)
All the normal properties of subtraction follow:
Go through the vector space problem provided here.
Question : Show that each of the conditions provided is in vector space
Solution: Conditions checked from the axioms and properties:
65 videos121 docs94 tests

1. What is a vector space? 
2. What are the key properties of a vector space? 
3. How do vector spaces relate to linear algebra? 
4. What is the dimension of a vector space? 
5. Can a vector space contain an infinite number of vectors? 

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