What do you understand by the term vector space?

Vector space can be imagined as a 3D space with a set of objects known as vectors, that can perform certain properties.

Answer: A vector space is a set of vectors that can be added together and multiplied by a scalar.

Go through the definition and various axioms related to vector spaces.

Explanation:

Definition:

A field is a set F of numbers with the property that if a, b ∈ F, then a + b, a − b, ab, and a / b are also in F (assuming, of course, that b not equal to 0 in the expression a / b)

Example:

We’re familiar with how to add, subtract, multiply, and divide the following sets of numbers

N = {0, 1, 2, 3, . . . }

Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . }

Q = na b | a, b ∈ Z, b not equal to 0

R = all real numbers

C = {a + bi | a, b ∈ R}

However, not all of these sets of numbers are fields of numbers. For example, 3 and 5 are in N, but 3 − 5 are not. Also, 3 and 5 are in Z, but 3 / 5 are not. This shows that N and Z are not fields of numbers. However, Q, R, and C are all fields of numbers

Axioms of Vector Spaces:

Vector can be defined using various axioms. Let x, y and z be the elements of vector space V and a & b be the elements of the field F.

1) Closed under Addition: For every element x, y in V, x + y is also in V

2) Closed under Scalar Multiplication: For every element x in V and scalar a in F, ax is in V

3) Commutativity of Addition:  For every element x, y in V, x + y = y + x

4) Existence of Multiplicative Identity: There exists an element in F notated as 1 so that for all x in v, 1x = x.

5) Existence of Additive Inverse:  For every element x in V, there exists another element in V such that x + (-x) = 0

6) Existence of Additive Identity

7) Associativity of Scalar Multiplication

8) Distribution of scalars to element

9) Distribution of elements to scalar

10) Associativity of Scalar Multiplication

Thus, a vector space is a set of vectors that can be added together and multiplied by a scalar.