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| [[LinearAlgebra/Elimination|Elimination]] and other factorizations offer a method for obtaining the '''particular solutions''' (''x,,p,,'') of a linear system. The '''complete solution''' (''x,,c,,'') is the generalized answer. | 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 == 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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| === First Example === | == Solution == |
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| The example used here [[LinearAlgebra/Elimination|here]]: {{{ x + 2y + z = 2 3x + 8y + z = 12 4y + z = 2 }}} Is rewritten as a linear system and eliminated into: {{{ ┌ ┐ ┌ ┐ ┌ ┐ │[1] 2 1 │ │ x│ │ 2│ │ 0 [2] -2 │ │ y│ = │ 6│ │ 0 0 [5]│ │ z│ │-10│ └ ┘ └ ┘ └ ┘ }}} This system solved to ''x=2, y=1, z=-2''. For a matrix with '''full column rank''', the system can usually be solved like this. === Second Example === Consider a system with '''free variables''': |
Consider the system: |
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| This system is rewritten as a linear system and eliminated into: | It was noted [[LinearAlgebra/ParticularSolution#Solutions|here]] that a particular solution is ''[-2 0 3/2 0]''. |
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| {{{ ┌ ┐ ┌ ┐ ┌ ┐ │[1] 2 2 2│ │ w│ │ 1│ │ 0 0 [2] 4│ │ x│ = │ 3│ │ 0 0 0 0│ │ y│ │ 0│ └ ┘ │ z│ └ ┘ └ ┘ }}} The columns without a pivot correspond to free variables. For any set of values in the free variable positions, there is some '''particular solution''' that satisfies the system. So aim for the simplest particular solution, where every free variable is set to 0. Then solve algebraically for the pivot variables: {{{ 2y + 4z = 3 2y + 4(0) = 3 2y = 3 y = 3/2 w + 2x + 2y + 2z = 1 w + 2(0) + 2y + 2(0) = 1 w + 2y = 1 w + 2(3/2) = 1 w + 3 = 1 w = -2 }}} ''w=-2, x=0, y=3/2, z=0'' is a solution. There is also the unstated [[LinearAlgebra/NullSpace#Zero_Vector_as_a_Solution|zero vector solution]] of ''w=0, x=0, y=0, z=0''. But there are also infinitely many ''other'' solutions making use of the free variables. ---- == Complete Solution == The '''complete solution''' is defined as ''x,,c,, = x,,p,, + x,,0,,''. That is, the [[LinearAlgebra/NullSpace#Solution|null space]] must be solved and added to the particular solution. The null space solutions are ''[-2 1 0 0]'' and ''[2 0 -1 1]'' for the second example above. '''''The''''' null space solution is any linear combination of the null space solution'''''s'''''. Again, for the second example above: |
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: |
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| ┌ ┐ ┌ ┐ ┌ ┐ │ -2 │ │ -2 │ │ 2 │ │ 0 │ + C │ 1 │ + C │ 0 │ │ 3/2│ 1│ 0 │ 2│ -1 │ │ 0 │ │ 0 │ │ 1 │ └ ┘ └ ┘ └ ┘ |
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 │ └ ┘ └ ┘ └ ┘ |
General Solution
A complete solution is a generalization of a particular solution. It is generally notated as xc.
Contents
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 null space must be solved and added to the 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.
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 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 │
└ ┘ └ ┘ └ ┘