= General Solution =

A '''complete solution''' is a generalization of a [[LinearAlgebra/ParticularSolution|particular solution]]. It is generally notated as ''x,,c,,''.

<<TableOfContents>>

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== Description ==

A consistent linear system has either one or infinitely many solutions. The general solution describes all of them.

A complete solution is formalized as ''x,,c,, = x,,p,, + x,,0,,''. That is, the [[LinearAlgebra/NullSpace#Solution|null space]] must be solved and added to the [[LinearAlgebra/ParticularSolution|particular solution]].

The key is that any combination of the null space vectors can be added to a particular solution and give a new particular solution, because they have an identity property.

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== Solution ==

Consider the system:

{{{
w + 2x + 2y + 2z = 1
2w + 4x + 6y + 8z = 5
3w + 6x + 8y + 10z = 6
}}}

It was noted [[LinearAlgebra/ParticularSolution#Solutions|here]] that a particular solution is ''[-2 0 3/2 0]''.

Furthermore, it was noted [[LinearAlgebra/NullSpace#Solutions|here]] that the null space vectors are ''[-2 1 0 0]'' and ''[2 0 -1 1]''.  The null space solution is ''any'' linear combination of these vectors. Consider:

{{{
       ┌    ┐     ┌    ┐
       │ -2 │     │  2 │
x  = C │  1 │ + C │  0 │
 0    1│  0 │    2│ -1 │
       │  0 │     │  1 │
       └    ┘     └    ┘
}}}

Altogether, the complete solution for the second example above is:

{{{
x  = x  + x
 c    p    0

     ┌    ┐     ┌    ┐     ┌    ┐
     │ -2 │     │ -2 │     │  2 │
x  = │  0 │ + C │  1 │ + C │  0 │
 c   │ 3/2│    1│  0 │    2│ -1 │
     │  0 │     │  0 │     │  1 │
     └    ┘     └    ┘     └    ┘
}}}



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