Website owner: James Miller
ALGEBRA OVER A FIELD
Algebra over a field. An algebra (or linear algebra) over a field F is a ring R that is also a vector space with members of F as scalars and satisfies (ax)(by) = (ab)(xy) for all scalars a and b and all members x and y of R. The dimension of the vector space is the order of R. The algebra is a commutative algebra, or an algebra with unit element , according as the ring is a commutative ring, or a ring with unit element. A division algebra is an algebra that is also a division ring. A simple algebra is an algebra that is a simple ring.
James & James. Mathematics Dictionary
Examples.
1. The set of real numbers is a commutative division ring over the field of rational numbers.
2. The set of all square matrices of order n with complex numbers (or real numbers) as elements is a non-commutative algebra with identity over the field of real numbers.
3. The space of all linear operators on a vector space, with composition as the product, is an algebra with identity.
References
James & James. Mathematics Dictionary
Hoffman, Kunze. Linear Algebra. p. 108
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
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
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
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
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
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
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