Website owner: James Miller
LINEAR TRANSFORMATIONS, LINEAR MAPPINGS, LINEAR OPERATORS ARE LINEAR VECTOR FUNCTIONS ASSIGNING OBJECTS TO OBJECTS
In courses in mathematical analysis one studies functions of one or more variables i.e. functions such as y = f(x), g = f(x,y), h = f(x,y,z), etc.. These functions can be regarded as functions of points or vectors in a space. For example, g = f(x,y) can be viewed as g = f (X) where X is a point or vector (x,y) in two dimensional space. Similarly h = f(x,y,z) can be viewed as h = f (X) where X is the point or vector (x,y,z) in three dimensional space. Thus we view them as functions of a vector argument. In the case of h = f (X) where X is the vector (x,y,z) the function assigns a number to the vector (x,y,z) in three dimensional space. This idea can be broadened to the idea of functions whose arguments may be, not just vectors in n-space, but other things – things like polynomials, continuous functions on some interval, infinite series, matrices, transformations, etc.. The idea can be broadened even further so that not only may the argument of the function be something other than a number or a vector in n-space but the value can also be something other than a number, also. Under the new, more abstract, definition a function has as its argument, “objects” and as its value, “objects” i.e. it assigns to the objects of one set objects of another set. Broadening the term “vector” to include objects of an arbitrary nature, we can define a vector function as a function that assigns a “vector” from one set to a “vector” of another set. We can then define a linear vector function (also known as a linear operator, a linear transformation and a linear mapping) as a function T: V→W, whose domain V is one vector space and range W is another vector space, which maps vectors from V into W in a linear way, mapping ax + by into ax' + by' for all a and b if it maps vectors x and y into x' and y'. The “vectors” of V and W can be any objects of an arbitrary nature for which addition and multiplication by a number make sense.
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?