= General Solution =

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

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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/Basis|basis]] of the [[LinearAlgebra/NullSpace#Solution|null space]] must be identified, and linear combinations of it must be added to the [[LinearAlgebra/ParticularSolution|particular solution]]. Since all such combinations evaluate to 0, 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 a [[LinearAlgebra/Basis|basis]] for the null space is ''{[-2 1 0 0], [2 0 -2 1]}''.  The null space solution is ''any'' linear combination of these vectors. Consider:

{{attachment:null.svg}}

Altogether, the complete solution for the second example above is ''x,,c,, = x,,p,, + x,,0,,''.

{{attachment:complete.svg}}

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== Row Space Solution ==

An alternative strategy follows from calculating a particular solution and the null space. Next, [[LinearAlgebra/Projection|project]] the particular solution onto the null space. The projection is notated ''x,,n,,''.

The given particular solution can be related to that projection as ''x,,p,, = x,,r,, + x,,n,,''. This should look similar to the formulation of a complete solution; there is a component spanning row space, and there is a another spanning null space. But the minimum [[LinearAlgebra/Norm|norm]] particular solution does not move through the null space at all. So, evaluate ''x,,r,, = x,,p,, - x,,n,,'' to arrive at the '''row space solution'''.

The optimized complete solution is ''x,,c,, = x,,r,, + c x,,n,,''.



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CategoryRicottone