Website owner: James Miller
Row space of a matrix, column space
Row space of a matrix. The row space of a matrix is that subspace spanned by the rows of the matrix (rows viewed as vectors). It is that space defined by all linear combinations of the rows of the matrix.
Consider a matrix containing five rows and three columns. The rows may be viewed as 3-vectors spanning some subspace of three-dimensional space. If the rows contain three linearly independent vectors they span all of three-dimensional space. If the rows contain only two linearly independent vectors they span the subspace of three-dimensional space defined by these two vectors (some plane passing through the origin). If all rows are multiples of some one row they represent one-dimensional subspace of three dimensional space corresponding to some line passing through the origin.
The effect of the elementary row operations on a matrix is to produce other sets of rows in the same row space. If the rows of a matrix A span some subspace K of n-space Vn then the elementary row operations will produce another matrix whose row vectors span the same subspace of Vn . Row-equivalent matrices have the same row space. The dimension of the row space corresponds to the number of linearly independent vectors required to span the row space — which is equal to the rank of the matrix.
Column space of a matrix. The column space of a matrix is the subspace spanned by the columns of the matrix (columns viewed as vectors).
Theorem. For any mxn matrix the dimension of its row space is equal to the dimension of its column space and both dimensions are equal to its rank. In other words the number of linearly independent rows in a matrix is equal to the number of linearly independent columns.
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