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 u, v∈ and vectors a and b,

  1. (distributivity): (u + v)a = ua + va
  2. (associativity): u(va) = (uv)a
  3. (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 (ab) and has the following properties:

  1. (commutativity): a.b = b.a
  2. (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.

(a + b).(a + b)= a.a + a.b + b.a + b.b
 = a.a + 2a.b + b.b

Two vectors whose scalar product is zero are called orthogonal or perpendicular. For example, the following pairs of 3-tuples are orthogonal: a and (0, 1, 0), (1, 0, 1) and (2, 1, -2). For orthogonal vectors we have the following generalization of the Pythagorean Theorem:

  (a + b).(a + b) = a.a + b.b

If we introduce the length (also called the norm) of vector a as ||a||2 = a.a, then the Pythagorean theorem admits a more conventional appearance:

  ||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?

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