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| = Solution = | = General Solution = |
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| [[LinearAlgebra/Elimination|Elimination]] is a fundamental step in solving systems of equations, but the method demonstrated on that page relies upon the right hand side being known values. This page instead supposes that the right hand side is unknown. | A '''complete solution''' is a generalization of a [[LinearAlgebra/ParticularSolution|particular solution]]. It is generally notated as ''x,,c,,''. |
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| == Introduction == | == Description == |
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| {{{ w + 2x + 2y + 2z = a 2w + 4x + 6y + 8z = b 3w + 6x + 8y + 10z = c }}} |
A consistent linear system has either one or infinitely many solutions. The '''general solution''' describes all of them. |
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| This corresponds to the following [[LinearAlgebra/Elimination|eliminated]] augmented matrix: {{{ ┌ ┐ │ [1] 2 2 2 a│ │ 0 0 [2] 4 b-2a│ │ 0 0 0 0 c-b-a│ └ ┘ }}} The immediate question then is: for what values of ''a'', ''b'', and ''c'' is this system solvable? |
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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| == Trivial Solvability == | == Solution == |
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| The above system can be easily shown to be solvable for some values. | Consider the system: |
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| 0w + 0x + 0y + 0z = c - b - a 0 = c - b - a |
w + 2x + 2y + 2z = 1 2w + 4x + 6y + 8z = 5 3w + 6x + 8y + 10z = 6 |
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| In other words, for any values of ''a'' and ''b'' which sum to ''c'', this system is solvable. | 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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| == General Solvability == | == Row Space Solution == |
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| ---- | 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,,''. |
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| 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'''. | |
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== Solutions == The first step to computing the '''complete solution''' (''x,,c,,'') is computing a '''particular solution''' (''x,,p,,''). Find a valid set of values for the right hand side. In this system, for example, [1 5 6]. Set all '''free varaibles''' to zero. Substitute to find all '''pivot variables'''. {{{ 2y = 3 y = 3/2 w + 2y = 1 w + 2(3/2) = 1 w + 3 = 1 w = -2 }}} The particular solution is: {{{ [-2 0 3/2 0] }}} Next, compute the [[LinearAlgebra/NullSpaces|null space]]. If the matrix is '''full rank''', the particular solution is the complete solution. Finally, the complete solution is the particular solution plus any combination of the null space. {{{ ┌ ┐ ┌ ┐ ┌ ┐ │ -2 │ │ -2 │ │ 2 │ │ 0 │ + C │ 1 │ + C │ 0 │ │ 3/2│ 1│ 0 │ 2│ -1 │ │ 0 │ │ 0 │ │ 1 │ └ ┘ └ ┘ └ ┘ }}} The values ''C,,1,,'' and ''C,,2,,'' are any number, because any combination of the two vectors is valid for the complete solution. |
The optimized complete solution is ''x,,c,, = x,,r,, + c x,,n,,''. |
General Solution
A complete solution is a generalization of a particular solution. It is generally notated as xc.
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 xc = xp + x0. That is, the basis of the null space must be identified, and linear combinations of it must be added to the particular solution. Since all such combinations evaluate to 0, they have an identity property.
Solution
Consider the system:
w + 2x + 2y + 2z = 1 2w + 4x + 6y + 8z = 5 3w + 6x + 8y + 10z = 6
It was noted here that a particular solution is [-2 0 3/2 0].
Furthermore, it was noted here that a 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:
Altogether, the complete solution for the second example above is xc = xp + x0.
Row Space Solution
An alternative strategy follows from calculating a particular solution and the null space. Next, project the particular solution onto the null space. The projection is notated xn.
The given particular solution can be related to that projection as xp = xr + xn. 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 norm particular solution does not move through the null space at all. So, evaluate xr = xp - xn to arrive at the row space solution.
The optimized complete solution is xc = xr + c xn.