Website owner: James Miller
Isomorphisms and homomorphisms effected by linear transformations from one abstract vector space into another
Let V be an n-dimensional and W be an m-dimensional abstract vector space over the field F. Suppose three linearly independent vectors spanning some subspace R of V are imaged into three linearly independent vectors spanning some subspace S of W under a linear transformation T. Then an isomorphism has been established between subspaces R in V and S in W. These two three dimensional subspaces are isomorphic to each other. Now suppose three linearly independent vectors spanning some subspace R in V are imaged into three vectors spanning some two dimensional subspace S of W under linear transformation T. Then a homomorphism has been established between subspaces R in V and S in W. These two subspaces are homomorphic to each other. If n linearly independent vectors in space V were imaged into n linearly independent vectors spanning some subspace S of W then vector space V would be isomorphic to subspace S of W. In general, if p linearly independent vectors spanning some subspace of V are imaged into k linearly independent vectors spanning some subspace of W then an isomorphism is established between the two subspaces if k = p and a homomorphism is established if k < p (the case k > p is impossible).
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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Principles for Living Life
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Theory on the Formation of Character
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Avoiding problems and trouble in life
Role of habit in formation of character
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Personal attributes of the true Christian
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Love of God and love of virtue are closely united
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Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
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Knowledge is one thing, wisdom is another
My views on Christianity in America
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The Prime Mover that decides "What We Are"
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Sin is serious business. The punishment for it is real. Hell is real.
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We are our habits
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