Website owner: James Miller
LINEAR SPACE (ABSTRACT VECTOR SPACE)
The definition of an abstract vector space is an obscure, abstruse, axiomatic one as are the definitions of many concepts of modern, abstract, axiom-based mathematics. A set of abstract algebraic laws or properties are stated as postulates that must be satisfied in order for the system to qualify as a vector space. These postulates derive from a list of properties of n-dimensional space.
Def. Linear Space (or Abstract Vector Space). A linear space is a set V of elements, called vectors, of an arbitrary nature for which the concepts of a sum and scalar multiplication (i.e. a product by a number) make sense, which is closed with respect to addition and scalar multiplication, and which satisfy the following postulates for any element x,y and z in V and any scalars a,b and c in a number field F:
1. (x+y) + z = x + (y+z)
2. There exists a "null" element 0 such that x+0 = x for every x in V
3. For every element x there exists an opposite -x such that x+(-x) = 0
4. x + y = y + x
5. 1x = x for the unit 1 in F and for all x
6. a(bx) = (ab)x
7. (a+b)x = ax + bx
8. c(x+y) = cx + cy
Postulates 1 through 4 are equivalent to the statement that V is an Abelian group with addition as the group operation.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?