Website owner:  James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

         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.

More from

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 Homosexuality

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

Moral Perversion

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?

[ Home ] [ Up ] [ Info ] [ Mail ]