Cut the knot: learn to enjoy mathematics
A math books store at a unique math study site. Learn to enjoy mathematics.
Google
Web CTK
Best sites for teachers
Sites for teachers
Sites for parents
Terms of use
Awards

Interactive Activities
CTK Exchange
CTK Insights - a blog

Games & Puzzles
What Is What
Arithmetic/Algebra
Geometry
Probability
Outline Mathematics
Make an Identity
Book Reviews
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
Visual Illusions
My Logo
Math Poll
Cut The Knot!
MSET99 Talk
Other Math sites
Front Page
Movie shortcuts
Personal info
Reciprocal links
Privacy Policy

Guest book
News sites

Recommend this site

Best sites for teachers
Sites for teachers
Sites for parents

Education & Parenting

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

Vector Spaces

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||.

Copyright © 1996-2008 Alexander Bogomolny

28777976Page copy protected against web site content infringement by Copyscape


Search:
Keywords:


Latest on CTK Exchange
Math
Posted by Laura
2 messages
06:56 AM, Apr-15-08

Divisibility rules - Jargon buste ...
Posted by Carolyn
2 messages
08:35 AM, Apr-04-08

drawing puzzle
Posted by martin gran
31 messages
06:53 PM, May-09-08

conway's game of life
Posted by frequency
0 messages
11:52 PM, May-12-08

Mistake on the page (an aside, Be ...
Posted by Max
4 messages
10:28 AM, Feb-28-08

Need details on a part of Proof o ...
Posted by Manuel S.
2 messages
05:24 PM, May-16-08

Josephus Flavius (correction)
Posted by David Turner
1 messages
09:42 AM, May-14-08