Vector Space

A vector space, also known as a linear space, is a collection of items known as vectors that are put together and multiplied (“scaled”) by numbers known as scalars. Scalars are commonly thought of being real numbers. With vector spaces, however, there are only a few cases of scalar multiplication by rational numbers, complex numbers, and so on. Vector addition and scalar multiplication algorithms must meet certain criteria, such as axioms. Scalars are defined as real or complex numbers using the phrases real vector space and complex vector space.

The associative and commutative laws of vector addition, as well as the associative and distributive processes of vector multiplication by scalars, define vector space. A vector space is made up of a set of V (named vectors by its elements), a field F (called scalars by its elements), and two operations.

• The action of vector addition takes two vectors, u, v ∈ V, and creates the third vector, u + v ∈ V.
• Scalar Multiplication is a mathematical operation that takes a scalar c ∈ F and a vector v ∈ V and returns a new vector cv ∈ V.

When both operations must meet the following criteria.

The elements of V are commonly referred to as vectors, while the elements of F are commonly referred to as scalars. The addition and multiplication operations must adhere to a set of axioms in order to qualify the vector space V. The axioms are a generalization of the vector attributes introduced in the field F. It’s called a real vector space if it’s over real numbers, and a complex vector space if it’s over complex numbers.

Difference between vector and vector space:-

A member of a vector space is a vector.

A vector space is a collection of items that may be multiplied by regular numbers and put together using axioms known as vector space axioms.

Basically, we can say that a vector is a component of vector space, which is a collection of objects multiplied by scalars and joined by the vector space axioms.

Equal Vectors:-

Equal vectors are defined as vectors that have the same magnitude and direction. The addressed line segments are parallel when two vectors are equal. Their vector columns are also the same.

Two vectors are identical if and only if their magnitudes are equal in the same direction. Vectors a and b are parallel and pointing in the same direction as shown in the diagram, but their magnitudes are not identical. As a result, we can deduce that the vectors in question are not equal.

Vector Space Axioms:-

All of the axioms should be quantified in some way. It should follow some of the axioms for vector addition and scalar multiplication. There are eight axiom rules listed below.

• Condition for Vector Addition:-

The following conditions must be met by an operation vector addition’+ ‘:

Closure: If x and y are any vectors in the vector space V, then x + y is also a member of V.

Commutative law: It states that x + y = y + x for all vectors x and y in V.

Associative Law:  any vectors x, y, and z in V, it states that x + (y + z) = (x + y) + z.

Additive Identity: The additive identity element is indicated by ’0’ in the vector space for every vector x in V, such that 0 + x = x = x + 0 .

Additive inverse: There is an additive inverse -x for each vector x in V to produce a solution in V.

• Condition for Scalar Multiplication:-

Between a scalar and a vector, a scalar multiplication operation is defined, and it must satisfy the following condition:

Closure: In the vector space V, if x is any vector and c is any real number, then x. c belongs to V.

Associative Law: For all real numbers c and d, and the vector x in V, it states that

1. (d. v) = (c. d). v

Distributive Law: For all real values c and d, and the vector x in V, this law states that (c + d). v = c.v + c.d

Unitary Law: This law states that

1.v = v.1 = v ;for all v in vector space V.

Properties of Vector Space:-

The following are some basic properties derived from the axioms:

• The addition operation of a finite list of vectors v1 v2,…, vk can be calculated in any order, and the result will be the same.
• If x + y = 0, then y = – x is a must.
• Negative of 0 is 0. This signifies that the value of 0 equals that of -0.
• The vector’s negation, or negative value of the vector’s negation, is the vector itself: -(-v) = v .
• x + y = x  , if and only if y = 0. As a result, the only vector that behaves like 0 is 0 itself.
• The zero vector is obtained by multiplying any vector by zero times. For each vector in y, 

 0 × y = 0.

Conclusion:-

A vector space (also known as a linear space) is a collection of vectors that may be added together and multiplied (“scaled”) by integers called scalars in mathematics, physics, and engineering. Scalar multiplication by complex numbers or, more broadly, by a scalar from any mathematical field is possible in various vector spaces. Vector addition and scalar multiplication must obey a set of rules known as vector axioms.