Vector Spaces

Introduction:

Vector spaces are a collection of objects called vectors that can be added to and multiplied by scalars according to certain criteria. Vector spaces appear in a variety of fields of mathematics as well as in a variety of applications; as a result, they are extremely important and useful.

Vector Space Definition: What is Vector Space?

Under finite vector addition and scalar multiplication, a vector space V is closed. It is a group of vectors that are added and then multiplied by a scalar quantity.

Thus the Vector Space Definition is therefore as follows: A vector space V is a collection of objects with a (vector) addition and scalar multiplication that are closed under both operations. The fundamental example is the n-dimensional Euclidean space Rn, in which each element is represented by a list of n real numbers, scalars are real numbers, addition is component-wise, and scalar multiplication is multiplication on each term independently.

When the scalars in a generic vector space are members of a field F, V is referred to as a vector space over F.

n-space Euclidean A real vector space is known as Rn, while a complex vector space is known as Cn.

The word “closed” mentioned above means that for all α, β ∈ F and x, y ∈ V αx + βy ∈ V (i.e. you can’t exit a vector space V using vector addition and scalar multiplication, it will not terminate, but  recurse). Also, when we write for α, β ∈ F (Field) and x ∈ V (Vector Space) (α + β)x the ‘+’ is in the field, whereas when we write x + y for x, y ∈ V , the ‘+’ is in the vector space. There is a double usage of this symbol.

Vector Space Axioms: What are vector Space Axioms and How are They to be Satisfied?

Vector space addition must satisfy the following vector space axioms:

(i)                   Distributive Law: (α + β)x = αx + βx for all x ∈ V and α, β ∈ F

(ii)                 Associative Law: α(βx)=(αβ)x

(iii)               Commutative Law:  x + y = y + x for all x, y ∈ V

(iv)              Associative Law:  x + (y + z)=(x + y) + z for all x, y, z ∈ V

(v)                 Distributive Law: α(x + y) = αx + αy

(vi)              ∃O ∈ V z 0 + x = x; 0 is usually called the origin

(vii)              Property of 0:  0x = 0

(viii)            Unitary Law: ex = x where e is the multiplicative unit in F

Properties of Vector Spaces:

Some vector space properties that are derived from the axioms are

• Every vector space has its own unique additive identity: Assuming a finite list of vectors v1 v2, . . , vk , whose sum can be calculated in any given order, the solution of the addition process will always be the same.

• If x + y = 0, then y = -x is the correct value (simple mathematical procedure).
• Negation cannot be applied to 0. As a result, the value of -0 is equal to zero.
• The negation of a vector, or negative value of the negation, is the vector itself: (v) = v.
• If x + y = x, a possibility stands that y = 0, then as a result, 0 is the only vector that has the same behaviour as 0/null vector. (Fact: a null vector has neither magnitude nor direction)
• A zero vector is obtained by multiplying any vector by zero times. For all vectors in y, 0 x y = 0
• A scalar multiple of any zero vector is the zero vector for every real number c. c0 = 0 is the value of the variable c0.
• If c.x=0, then either c or x must be zero. The product of a scalar and a vector is equal to 0 when either scalar is 0 or a vector is of 0 value.

• Multiplying the scalar value −1 times with a vector is ultimately the negation of the vector: (−1)x = −x. We can define the subtraction in terms of addition by defining x − y as an abbreviation for x + (−y).

Conclusion:

In a type of geometrical space such as Euclidean space, its elements are vectors, and we have the concepts of vector length and distance between points. A scalar product of two vectors is also a concept mentioned above. We may define limits and continuity using these principles, bridging the gap between algebra and analysis. Vector spaces are used in mathematics, science, and engineering today. To deal with systems of linear equations, they are the suitable linear-algebraic notion.Vector spaces also provide an abstract, coordinate-free approach of working with geometrical and physical objects like tensors.