Definitions:Ultimate Space = S
si = arbitrary partition of S
Δsi = infinitesimal si, that is limit of si as it tends to zero.
pi = arbitrary point in S = Δsi
Problem 1: si ⊂ S implies si is contained in S. True or false?
Problem 2: ∑∞si = S, that is, lim si = S as i tends to infinity. True or False?
Ans: true. All problem 2 is saying is that the ultimate space S = the sum of its parts.
Problem 3: a continuum formed by the arrangement of pis in the same direction forms a straight line. Describe the arrangement of pis that:
(a) forms a horizontal continuum
(b) forms a vertical continuum
Ans: (i) west-east or east-west arrangement
(ii)north-south or south-north arrangement.
Problem 4 : The continuum of pi is a curve if it is not a straight line. That is, all the points are not arranged in the same direction. The continuum is closed if it ends where it started. suppose a curve C encompasses the space si:
(i) What is C relative to si?
(ii) What are the points on C called?
(iii) What are the points in si called?
Ans: (i) C is the boundary of si
(ii) The points on C are the boundary-points of si
(ii) The points in si are the interior points of si.
Problem 5 : what properties must si have to be a Linear space over a field F (usually the Real number field or the Complex number field)?
Ans: the space si must have the following properties to be a Linear space:
(a) The result of the addition of any two elements in si must be an element in si
(b) The result of a scalar multiplication of the field F and elements of si must be an element in si.
Additional properties for any x, y, z in si and α, β in F are:
(c) (x + y) + z = x +(y + z)
(d) x + y = y + x
(e) 0 is in si such that x + 0 =x for all x in si
(f) For each x in si, there is an element -x (the additive inverse of x) in si
such that x + (-x) = 0
(g) α(x + y) = αx + αy
(h) (α + β)x = αx + βy
(i) α(βx) = (αβ)x
(j) 1(x) = x.
The elements of a linear space are called points or vectors and the elements of the field are called scalars. Usually, the field is the real number field or the complex number field.
A linear space is also called a vector space. An example is the vector space Vn(R), the n implies that the vectors are n-tuples of real numbers. The scalars are real numbers with addition and scalar multiplication defined as follows:
(α1,...,αn) + (β1,...,βn) = (α1 + β1,...,αn + βn)
γ(α1,...,αn) = (γα1,...,γαn)
Problem 6: what do the properties in ans 5(a) thru 5(f) tell us about a linear space? Ans: the properties in ans 5(a) thru 5(f) tell us that a linear space is an Abelian group.
Problem 7: the vector space Vn(R) is a linear space. What other linear space do you know?
Ans: the set X of all functions from a non empty set T into a field F such that addition and scalar multiplication is defined as follows:
[f + g](t) = f(t) + g(t) and [αf](t) = αf(t), where, f, g are in X, t in T and α in F.
An m x n matrix is also a linear space.
Problem 8: what is the diferrence between a subspace of a linear space and a linear manifold?
Ans: no difference. a linear subspace is also called a linear manifold.
Problem 9: what is a quotient space?
Ans: The linear space X/E is called the quotient space of X modulo E. X is a linear space and E its subspace.
Problem 10: Given that n = 1 in the vector space Vn(R), what type of space is V1(R)?
Ans: The real number line. Consequently, the real number line is a linear space over itself. In essence a 1-dimensional linear space.
Peter Oye Sagay