Website owner: James Miller
PROPERTIES THAT FOLLOW FROM THE AXIOMATIC DEFINITIONS OF DIFFERENT SPACES
In modern abstract mathematics properties are deduced from sets of axioms just as in Euclidean Geometry all results are arrived at through deduction from a set of axioms.
1. Linear Space. The following properties follow directly from the eight axioms that constitute the definition of a linear space:
- subspaces (the space may contain subspaces)
- basis vectors (if the space contains subspaces there will be
basis vectors that span the subspaces such that any vector in
a subspace can be expressed as a linear combination of a set
of basis vectors)
- dimension (the space has a dimension given by the number of
linearly independent vectors required to span it)
- coordinate representation of vectors (if e1, e2, .... , en are a set
of independent basis vectors for the space any vector v in the
space can be represented as
v = a1e1 + a2e2 + .... + anen
where a1, a2, .... , an are numbers from a number field F
which can be regarded as coordinates of the vector v with
respect to that basis)
2. Linear space with an inner product. A linear space with an inner product has all the properties of a linear space plus the following properties:
- orthogonality (concept of the orthogonality of one vector to
- orthogonal bases
3. Linear space with an inner product and a norm. A linear space with both an inner product and a norm has all the properties of a linear space with an inner product plus:
- orthonormal bases
- coordinate representation with respect to orthonormal bases
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?