SolitaryRoad.com

Website owner: James Miller

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

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 x_{p} be a particular solution to the system Ax = b i.e. vector x_{p} can be any
vector that satisfies the system. Then the complete solution of the system Ax = b can be written
as

X = x_{p} + c_{1}u_{1} + c_{2}u_{2} + .... + c_{n-r}u_{n-r}

where c_{1}, c_{2}, .... ,c_{n-r} are arbitrary constants and the vectors u_{1}, u_{2}, .... ,u_{n-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 u_{1}, u_{2}, .... ,u_{n-r} which
satisfy the set of homogeneous equations Ax = 0. The vector x_{p} plus any linear combination of
these vectors u_{1}, u_{2}, .... ,u_{n-r} is a solution of the given equation. There are no other solutions. If b
= 0, the vector x_{p} can be taken as x_{p} = 0.

Example. Solve the system:

We divide the first equation by 4 and use it to eliminate x_{1} 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 x_{3} 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 x_{2} and x_{4}. Let us choose
x_{2} = x_{4} = 0. Substituting these values into the system we get

x_{1} = 13/9

x_{3} = -2/9

Thus the particular solution is:

To find two linearly independent solution vectors, we take the homogeneous equation

and choose arbitrary values for x_{2} and x_{4}.

For the first vector let us take x_{2} = 0, x_{4} = 3. On substituting into the system we get

+ 1/3 = 0

- 5/3 = 0

which has the solution

x_{1} = -1/3, x_{3} = 5/3 .

Thus the first vector is:

For the second vector let us pick the values x_{2} = 3, x_{4} = 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 c_{1} and c_{2} are arbitrary constants.

More from SolitaryRoad.com:

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

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