Home

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Ultimate Space = S

s_{i} = arbitrary partition of S

Δs_{i} = infinitesimal s_{i}, that is limit of s_{i} as it tends to zero.

p_{i} = arbitrary point in S = Δs_{i}

**Problem 1**: s_{i} ⊂ S implies s_{i} is contained in S. True or false?
**Ans: true.**

**Problem 2**: ∑^{∞}s_{i} = S, that is, lim s_{i} = 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 p_{i}s in the same direction forms a straight line. Describe the arrangement of p_{i}s 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 p_{i} 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 s_{i}:

(i) What is C relative to s_{i}?

(ii) What are the points on C called?

(iii) What are the points in s_{i} called?
**Ans: (i) C is the boundary of s _{i}
(ii) The points on C are the boundary-points of s_{i}
(ii) The points in s_{i} are the interior points of s_{i}. **

**Problem 5 **: what properties must s_{i} have to be a Linear space over a field F (usually the Real number field or the Complex number field)?
**Ans: the space s _{i} must have the following properties to be a Linear space:
(a) The result of the addition of any two elements in s_{i} must be an element in s_{i}
(b) The result of a scalar multiplication of the field F and elements of s_{i} must be an element in s_{i}.
Additional properties for any x, y, z in s_{i} and α, β in F are:
(c) (x + y) + z = x +(y + z)
(d) x + y = y + x
(e) 0 is in s_{i} such that x + 0 =x for all x in s_{i}
(f) For each x in s_{i}, there is an element -x (the additive inverse of x) in s_{i}
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 V_{n}(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:
(α^{1},...,α^{n}) + (β^{1},...,β^{n}) = (α^{1} + β^{1},...,α^{n} + β^{n})
γ(α^{1},...,α^{n}) = (γα^{1},...,γα^{n})
**

**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 V_{n}(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 V_{n}(R), what type of space is V_{1}(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