Website owner: James Miller
Solution of a consistent system of linear equations
Solution of a consistent system of linear equations. The following theorem gives the complete solution of a consistent system of linear equations Ax = b. It deals not only with the case in which the rank of the matrix A is equal to the number of unknowns, in which case there is one single solution. It also deals with the case of underdetermined systems where the rank of A is less than the number of unknowns. The complete solution to the system in this case consists of any particular solution of the system Ax = b plus all solutions of the system Ax = 0 (a linear manifold of solutions).
Theorem. Let Ax = b be a consistent system having n unknowns, and let the rank of A be r.
Case 1. r = n. There is a single solution vector x which can be found by one of the usual methods.
Case 2. r < n. Let xp be a particular solution to the system Ax = b i.e. vector xp can be any vector that satisfies the system. Then the complete solution of the system Ax = b can be written as
X = xp + c1u1 + c2u2 + .... + cn-run-r
where c1, c2, .... ,cn-r are arbitrary constants and the vectors u1, u2, .... ,un-r are any set of linearly independent vectors that span the solution space of the system Ax = 0. There are n-r such vectors. In other words, one can find n-r linearly independent vectors u1, u2, .... ,un-r which satisfy the set of homogeneous equations Ax = 0. The vector xp plus any linear combination of these vectors u1, u2, .... ,un-r is a solution of the given equation. There are no other solutions. If b = 0, the vector xp can be taken as xp = 0.
Example. Solve the system:
We divide the first equation by 4 and use it to eliminate x1 from the remaining equations:
Now we rearrange to put the largest element in the proper position:
We divide the second equation by 4.5 and use it to eliminate x3 from the other two equations:
We see that the rank of A is 2 and that the system has two equations.
Because there are four unknowns and the rank of A is 2, we know from the theorem that the complete solution is made up of a particular solution and any linear combination of two linearly independent solution vectors.
We can find the complete solution as follows:
Add .25 times row 2 to row 1, obtaining:
We can find the particular solution by choosing two arbitrary values for x2 and x4. Let us choose x2 = x4 = 0. Substituting these values into the system we get
x1 = 13/9
x3 = -2/9
Thus the particular solution is:
To find two linearly independent solution vectors, we take the homogeneous equation
and choose arbitrary values for x2 and x4.
For the first vector let us take x2 = 0, x4 = 3. On substituting into the system we get
+ 1/3 = 0
- 5/3 = 0
which has the solution
x1 = -1/3, x3 = 5/3 .
Thus the first vector is:
For the second vector let us pick the values x2 = 3, x4 = 0. On substituting into the system we get:
+ 1 = 0
- 2 = 0
which has the solution
= -1, = 2 .
The second vector is then
and the general solution is:
where c1 and c2 are arbitrary constants.
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 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
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?