SolitaryRoad.com

Website owner:  James Miller


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

The solution set of the linear system ax = 0 is a vector space


Consider the system of m linear equations in n unknowns x1, x2, .... ,xn


             ole.gif


or, more concisely, AX = 0. Let the rank of the coefficient matrix A be r. If r = n the solution consists of only the single solution X = 0, which is called the trivial solution. If r < n there are an infinite number of solution vectors which will satisfy the system corresponding to all points in some subspace of n-dimensional space. To illustrate this let us consider a couple of simple examples from ordinary three-dimensional space.


Suppose the system AX = 0 consists of the single equation


             5x + 3y + 9z = 0 .


This equation corresponds to a plane in three-dimensional space that passes through the origin of the coordinate system. Any point on this plane satisfies the equation and is thus a solution to our system AX = 0. The set of all solutions to our system AX = 0 corresponds to all points on this plane. Furthermore, since the plane passes through the origin of the coordinate system, the plane represents a vector space. Why? Because a linear combination of any two vectors in the plane is also in the plane and any vector in the plane can be obtained as a linear combination of any two basis vectors in the plane. So, in summary, in this particular example the solution set to our system AX = 0 corresponds to the two-dimensional subspace of three-dimensional space represented by this plane. We call this subspace the solution space of the system AX = 0.


Let us consider another example. Suppose the system AX = 0 consists of the following two equations

     

                  2x + 5y + 3z = 0

                  9x - 2y - 8z = 0


These two equations correspond to two planes in three-dimensional space that intersect in some line which passes through the origin of the coordinate system. Any point of this line of intersection satisfies the system and is thus a solution to our system AX = 0. Furthermore, since the line passes through the origin of the coordinate system, the line represents a vector space. A linear combination of any two vectors in the line is also in the line and any vector in the line can be obtained as a linear combination of any basis vector for the line. So, in summary, in this example the solution set to our system AX = 0 corresponds to a one-dimensional subspace of three-dimensional space represented by this line of intersection of the two planes. In this case the solution space of the system AX = 0 is one-dimensional.


What determines the dimension of the solution space of the system AX = 0? The dimension is given by n - r. In our first example the number of unknowns, n, is 3 and the rank, r, is 1 so the dimension of the solution space was 3 - 1 = 2. In our second example n = 3 and r = 2 so the dimension of the solution space was 3 - 2 = 1.



Theorem. The solution set of the linear system AX = 0 is a vector space. This means that if X1 and X2 are any two solution vectors of AX = 0 and k1 and k2 are arbitrary constants then


                       X = k1X1 + k2X2


is also a solution vector of the system.


Proof. Since X1 and X2 are solutions, AX1 = 0 and AX2 = 0. Thus


            AX = A(k1X1 + k2X2) = k1AX1 + k2AX2 = k1·0 + k2·0 = 0 .



If r < n the solution space of AX = 0 is multi-dimensional of dimension n - r. If it has a dimension of s there will exist s linearly independent basis vectors that span the space. Any solution of the system can be written as some linear combination of these basis vectors. In general, the solution space of AX = 0 is some s-dimensional subspace of n-space. In 3-space it consists of some line or plane that passes through the origin of the coordinate system. Now let us note that although the solution set of AX = 0 does constitute a vector space the solution set of AX = B does not constitute a vector space. It is not true that if X1 and X2 are two solution vectors of AX = B and k1 and k2 are arbitrary constants that


                     X = k1X1 + k2X2


is a solution of the system. It is true that the solution set of AX = B may correspond to some line or plane of points in n-dimensional space but the line or plane doesn't pass through the origin of the coordinate system and does not represent a vector space. Only those lines or planes that pass through the origin represent vector spaces. Only they represent vector subspaces of n-dimensional space -- lines and planes not passing through the origin do not.


The solution space of the linear system AX = 0 is called the null space of matrix A. It is called this because if we view matrix A as a linear operator it images all points of this solution space into the null vector "0".




More from SolitaryRoad.com:

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

Television

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-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


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