General Solutions to a Linear System

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  • #1
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Main Question or Discussion Point

According to this,
http://upload.wikimedia.org/math/5/8/2/582b41cd88284564c46e711cbbca55fa.png
http://en.wikipedia.org/wiki/System_of_linear_equations#Homogeneous_systems

A general solution to some linear system is
a particular solution (P) + a homogeneous solution (H)

My question:
is this also true for any multiple of the particular solution +/ any multiple of the homogenous solution? Or does it have to be P + H or P-2H, etc.?

Thank you!
 

Answers and Replies

  • #2
AlephZero
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That Wiki page is badly written (ever heard the saying "free stuff is often work exactly what you paid for it"?)

The homogeneous solution can be more general than just scalar multiples of a single solution. It can be any scalar multiples of several independent solutions.

For example consider the equation system (1 equation with 3 variables)
x + 2y + 3z = 6

The homogeneous equation is
x + 2y + 3z = 0

Two independent homogeneous solutions of this are
x = 2, y = -1, z = 0
x = 3, y = 0, z = -1

They are called "independent" because one of them is NOT a multiple of the other.

The general homogenous solution is the sum of multiples of each independent solution, which is
x = 2A + 3B, y = -A, z = - B
where A and B are two arbitrary constants.

A particular solution of
x + 2y + 3z = 6
is x = 1, y = 1, z = 1

You can't muliply the particular solution by 2 (or any other factor), because
1 + 2.1 + 3.1 = 6
but
2 + 2.2 + 3.2 is not equal to 6.

The general solution is therefore
x = 1 + 2A + 3B, y = 1 - A, z = 1 - B

Note, the particular solution is not unique, but the difference between two particular solutions is always a multiple of the homogeneous solution.

For example
x = 6, y = 0, z = 0 is another particular solution.

The difference between that and
x = 1, y = 1, z = 1
is
x = 5, y = -1, z = -1
which is equal to the homogeneous solution when A = 1 and B = 1.

So, the solution sets written using different particular solutions might appear to be different, but they are really the same because they both represent the same infinite set of solutions to the equations.
 
  • #3
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That Wiki page is badly written (ever heard the saying "free stuff is often work exactly what you paid for it"?)

The homogeneous solution can be more general than just scalar multiples of a single solution. It can be any scalar multiples of several independent solutions.

For example consider the equation system (1 equation with 3 variables)
x + 2y + 3z = 6

The homogeneous equation is
x + 2y + 3z = 0

Two independent homogeneous solutions of this are
x = 2, y = -1, z = 0
x = 3, y = 0, z = -1

They are called "independent" because one of them is NOT a multiple of the other.

The general homogenous solution is the sum of multiples of each independent solution, which is
x = 2A + 3B, y = -A, z = - B
where A and B are two arbitrary constants.

A particular solution of
x + 2y + 3z = 6
is x = 1, y = 1, z = 1

You can't muliply the particular solution by 2 (or any other factor), because
1 + 2.1 + 3.1 = 6
but
2 + 2.2 + 3.2 is not equal to 6.

The general solution is therefore
x = 1 + 2A + 3B, y = 1 - A, z = 1 - B

Note, the particular solution is not unique, but the difference between two particular solutions is always a multiple of the homogeneous solution.

For example
x = 6, y = 0, z = 0 is another particular solution.

The difference between that and
x = 1, y = 1, z = 1
is
x = 5, y = -1, z = -1
which is equal to the homogeneous solution when A = 1 and B = 1.

So, the solution sets written using different particular solutions might appear to be different, but they are really the same because they both represent the same infinite set of solutions to the equations.
Thank you for the reply. So, if P is a solution to the original (or some point), and H is a homogeneous solution, the general set of solutions could be written as P + H , but not something like P-H or P+2H?
 
  • #4
HallsofIvy
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If H is the general solution, including undetermined constants, to the homogeneous equations, then the general solution to the non-homogeneous equations is of the form P+ H. Because you could always include "-1" or "2" in the coefficients for the general solution, that includes "P- H" and "P+ 2H".
 

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