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Concept of an operator. Examples of linear operators. Integral operator. Bounded, adjoint operators.
Concept of an operator. The term “operator” is another term for function, mapping or transformation. An operator assigns an object from one set (the co-domain) to an object from another set (the domain). If we are talking about vector spaces we think of an operator as “operating” on one vector to produce another vector. It is viewed as a kind of black box that operates on vectors to produce other vectors. The black box has an input and an output. We input a vector into the box and it then outputs a vector. An example is the matrix A in the matrix equation y = Ax where A is viewed as a black box that operates on the vector x to produce vector y. Here matrix A maps a vector x from one space (the domain) into the vector y in another space (the co-domain). If we are talking about functional spaces where a function f(x) is viewed as a vector, an operator is viewed as a black box that operates on a function to produce another function. The input to the box is a function and the output is another function.
which associates with function f a certain function g (where K(x, t) is a definite known function called the kernel).
Def. Linear operator. Let an operator A be defined on a vector space. It is linear if
A(av1 + bv2) = aAv1 + bAv2
for all vectors v1 and v2 and scalars a, b.
Examples of linear operators (or linear mappings, transformations, etc.) .
1. The mapping y = Ax where A is an mxn matrix, x is an n-vector and y is an m-vector. This represents a linear mapping from n-space into m-space.
2. The mapping y = Ax where A is an nxn matrix, x is an n-vector and y is an n-vector. This represent a linear mapping from n-space into n-space.
3. Let V be the vector space of all n-square matrices over the real field R and A be an n-square matrix over R.. Let v be any member of V. The transformation
w = Av
where w V, constitutes a linear transformation T: V → V.
4. Let V be the vector space of all polynomials in the variable x over the real field R. Then the derivative defines a linear transformation D: V → V where, for any polynomial f V, we let D(f) = df/dx. For example, D(4x2 + 3x + 5) = 8x + 3.
5. Let V be the vector space of all polynomials in the variable x over the real field R. Then the integral from, say, 0 to 1, defines a linear transformation T: V → R where, for any polynomial
f V, we let
6. Let V be the vector space of all real-valued continuous functions defined on the interval [0,1]. Then for any f V the transformation
defines a linear transformation T : V → V.
7. Let V be the vector space of all real-valued continuously differentiable functions defined on the interval [0,1] and W be the vector space of all real-valued continuous functions defined on the interval [0,1]. Then for any f V the transformation
D(f) = f '(x)
defines a linear transformation D:V → W.
8. The integral operator
which associates with function f a certain function g (where k(x, y) is a definite known function called the kernel).
Def. Integral operator. A linear operator that associates with every function f another function g by means of an integral equation.
Example. The formula
associates with every function f a certain function g. Symbolically, we can write this transformation as
g = Af
where the operator A is called an integral operator. For each vector f that A is provided, it gives a vector g. We can view it as a mapping where A maps each vector f into a vector g.
Def. Bounded operator. An operator A is called bounded if a positive real number C exists such that
|| Av || < C ||v||
Theorem. Let A be a bounded linear operator which maps a Hilbert space H into H (with the domain of A equal to H). Then a uniquely determined bounded linear operator A*, called the adjoint of A, exists such that the inner products (Af, g) and (f, A*g) are equal for all f, g ε H. If A = A*, the operator is called self-adjoint.
James & James. Mathematics Dictionary
Mathematics, Its Content, Methods and Meaning. Vol. 3, Chap. XIX
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
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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
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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?