Vector Theroy

Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Malinowski space. Gradient vector, the vector giving the magnitude and direction of maximum increase of a scalar field. Gyro-vector, a hyperbolic geometry version of a vector.

The topics dealt with in this introductory chapter are of a general mathematics-ical nature, being just as relevant to other parts of mathematics as they are to vector space theory. In this course you will be expected to learn several things about vector spaces (of course!), but, perhaps even more importantly, you will be expected to acquire the ability to think clearly and express your-

self clearly, for this is what mathematics is really all about. Accordingly, you are urged to read (or reread) Chapter 1 of “Proofs and Problems in Calculus” by G. P. Monro; some of the points made there are reiterated below.
ogic and common sense

When reading or writing mathematics you should always remember that the

mathematical symbols which are used are simply abbreviations for words.

Mechanically replacing the symbols by the words they represent should result in grammatically correct and complete sentences. The meanings of a few commonly used symbols are given in the following table.

Symbols To be read as 

                     {...|...} the set of all ...such that...

                         =  is 

                          ∈ is nor  in

                         > greater than  or is greater 

                               than 

Thus, for example, the following sequence of symbols                              

                                    {x∈X| a} 6= ∅ 

is an abbreviated way of writing the sentence The set of all x n X such that xis greater than axis not the empty set.