Vector Space and Spaces with the Scalar Products
An abstract real vector space is an commutative group with
one additional operation: its elements may be multiplied by real numbers (scalars). It's by no means a
group operation (except for the case when we look at the set R of real numbers as a real vector space)
because in a group operations both operands must come from the same set. Multiplication by a scalar
is required to satisfy three additional laws: for
- (distributivity): (u + v)a = ua + va
- (associativity): u(va) = (uv)a
- (distributivity): u(a + b) = ua + ub
These are variants of distributive and associative laws. As, an example, for n-tuple spaces, the multiplication by a scalar is defined componentwise:
u(a1, a2, a3) = (ua1, ua2, ua2) |
With this definition and addition defined also componentwise the set of 3-tuples becomes a vector space. It's important to understand that an n-tuple is only then is regarded as a vector when it's considered an element of a set where two operations (addition and multiplication by a scalar) are defined. Thus, vectors and Vector Spaces are born simultaneously.

Spaces with Scalar Product
For some vector spaces it's possible to define another multiplication - a scalar (or inner, or dot) product. The scalar product is defined for two vector operands with the result being a scalar. Therefore, the scalar product too is not a group operation. The scalar product of two vectors a and b is denoted a.b or (a, b) and has the following properties:
- (commutativity): a.b = b.a
- (distributivity): a.(b + c) = a.b + a.c
As an example, the scalar product for 3-tuples is defined in the following manner:
(a1, a2, a3).(b1, b2, b3) = a1b1 + a2b2 + a3b3 |
As an application of these laws, let's prove a simple but interesting identity.
(*) |
|
Two vectors whose scalar product is zero are called orthogonal or perpendicular. For example, the following pairs of 3-tuples are orthogonal:
a and
(a + b).(a + b) = a.a + b.b |
If we introduce the length (also called the norm) of vector a as
||a - b||2 = ||a||2 + ||b||2. |
Identity (*) is a generalization of the Cosine Law. In fact (*) is one of the reasons that the angle between two vectors is defined by:
cos(α) = a.b / ||a|| ||b||. |

What Can Be Multiplied?
- What Is Multiplication?
- Multiplication of Equations
- Multiplication of Functions
- Multiplication of Matrices
- Multiplication of Numbers
- Peg Solitaire and Group Theory
- Multiplication of Permutations
- Multiplication of Sets
- Multiplication of Vectors
- Multiplication of a Vector by a Matrix
- Vector Space and Spaces with the Scalar Product
- Addition and Multiplication Tables in Various Bases
- Multiplication of Points on a Circle
- Multiplication of Points on an Ellipse

|Contact| |Front page| |Contents| |Up|
Copyright © 1996-2018 Alexander Bogomolny72533663