Vector Space

Vector Space Definition

Before we ask ourselves to define vector space, there are few basic terms that we need to know in order to understand vector space perfectly. 

  1. Set: A set is a collection of distinct objects that are called elements.

  2. Elements: Elements are basically real or complex numbers which are used in mathematics. Elements of a set can also be a list of things. We usually notate a set by encasing the elements within curly braces. An element cannot be repeated within the same set.

  3. Space: In Mathematics, a space is a set in which the list of elements are elucidated by a collection of guidelines or axioms regarding how each element relates to another within the set.

  4. Axioms: Vector space axioms can be defined as  the operations of vector addition and scalar multiplication that must satisfy certain requirements.


Now that we are Fully Acquainted with Few Important Terms, we can now Move on to Vector Space. So, how do we Define Vector Space?


Vector space is a set of multidimensional quantities that are called vectors parallely with a set of one-dimensional quantities that are called scalars. 


What is Vector and Vector Space?

The vectors can be added as well as multiplied by scalars while preserving the ordinary arithmetic properties.


So, How do you Define a Vector Space?

A vector space is one in which the elements are sets of numbers themselves. Every element in a vector space is a list of objects with specific length, which we call vectors. The elements of a vector space are often referred to as n-tuples, where n is the specific length of each of the elements in the set. Matrix is another way of representing each element of a vector space of length n. 


History

Historically, the first ideas relating to vector spaces came from analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Analytic geometry was founded by  French Mathematicians René Descartes and Pierre de Fermat around 1636. They identified a solution to an equation of two variables with points on a plane curve. Bolzano introduced certain operations on points, lines and planes, which are called predecessors of vectors in order to achieve geometric solutions without using coordinates. The modern and more abstract treatment was formulated by Giuseppe Peano in 1888.


Vector spaces branches out from affine geometry, through the introduction of coordinates in the plane or three-dimensional space. In other words, vector spaces are mathematical objects. They abstractly capture the geometry and algebra of linear equations and are the central objects of study in linear algebra. They often appear throughout mathematics and physics. 


Vector Addition

Vector addition is a way of combining two vectors, say u and v, into a single vector like this: u+v.


There are few conditions that must be satisfied by the operation of vector addition. They are:

  1. Closure: If u and v are the vectors in V, the sum of u and v ( u+v) will belong to V.

  2. Commutative Law: For all the vectors (u and v) in V,  u + v is equal to  v + u.

  3. Associative Law: Vectors u, v, w in V,  u + (v + w) is equal to (u + v) + w.

  4. Additive Identity: The set V has an additive identity element which is usually denoted by 0, such that for any vector (v) in V, 0 + v = v and  v + 0 = v.

  5. Additive Inverses: For every vector v in V,  the equations v + x = 0  and x + v = 0 contains a solution x in V which is called an additive inverse of v, and is denoted by - v.


Scalar Multiplication

Scalar multiplication is a way of combining a scalar k, along with a vector v, to end up with the vector kv. The operation of scalar multiplication can be explained between real numbers and vectors that must satisfy few conditions, they are:

1) Closure: If v is a vector in V and c in real numbers, the product c-v will belong to V. 

2) Distributive Law: For the real number c and vectors u & v in V, c · (u + v) = c · u + c · v.

3) Distributive Law:  For the real numbers c & d and vectors v in V, (c+d) · v = c · v + d · v

4)  Associative Law: For the real numbers c & d and vectors v in V, c · (d · v) = (cd) · v

5) Unitary Law: For the vector v in V, 1 · v = v


Axioms for Vector Spaces

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

  1. Closed Under Addition: For every element x and 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 and y in V, x + y = y + x.

  4. Associativity of Addition: For every element x, y, and z in V, (x + y) + z = x + (y + z).

  5. Existence of the Additive Identity: There exists an element in V which is denoted as 0 such that x + 0 = x, for all x in V.

  6. Existence of the Additive Inverse: For every element x in V, there exists another element in V that we can call -x such that x + (-x) = 0.

  7. Existence of the Multiplicative Identity: There exists an element in F notated as 1 so that for all x in V, 1x = x.

  8. Associativity of Scalar Multiplication: For every element x in V, and for each pair of elements a and b in F, (ab)x = a(bx).

  9. Distribution of Elements to Scalars: For every element a in F and every pair of elements x and y in V, a(x + y) = ax + ay.

  10. Distribution of Scalars to Elements: For every element x in V, and every pair of elements a and b in F, (a + b)x = ax + bx.


Vector Space Examples

Here are the spaces of n-tuples where each part of every element is a real number. The set of scalars are also the set of real numbers. Let's take a look at some key definitions. 

  • Addition is explained as adding the corresponding parts of each element: (a, b, . . . ) + (c, d, .. . ) is equal to (a + c, b + d, .. .).

  • Scalar multiplication is explained as multiplying every part of the element by the scalar: a(b, c, . . . ) = (ab, ac, . . . ).

  • For these vector spaces the additive identity is the element (0, 0, 0, . . . , 0), where there are n 0s in this element.

  • For these vector spaces the multiplicative identity is the scalar 1 from the field of real numbers R.

The following are the basic vector space examples, but there is no proof that the space R3 is a vector space.

FAQ (Frequently Asked Questions)

Question 1) Are Vector “Arrows” that have a Direction & Magnitude?

Answer 1) Yes, most of the frequently-encountered vector spaces do have an arrow as their set of vectors, column matrices of real numbers like 2/1.


One of the popular ways to illustrate this column matrix is that we can draw an arrow that starts at the origin in the plane and will end at the point (2, 1). The idea of  “vectors are arrows” came from this unique and specific way to visualize the vectors of 1 specific example of a vector space. And for other vector spaces where the vectors are functions or infinite sequences, we can’t visualize vectors as arrows.

Question 2) Are the Real Numbers a Vector Space?

Answer 2) Yes, a set of vector space is always a vector space.