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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. [[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.
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=== First Example ===

The example used here [[LinearAlgebra/Elimination|here]]:
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w + 2x + 2y + 2z = a
2w + 4x + 6y + 8z = b
3w + 6x + 8y + 10z = c		 x + 2y + z = 2
3x + 8y + z = 12
4y + z = 2
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This corresponds to the following [[LinearAlgebra/Elimination|eliminated]] augmented matrix: Is rewritten as a linear system and eliminated into:
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┌   ┐
│ [1] 2 2 2 a│
│ 0 0 [2] 4 b-2a│
│ 0  0 0 0 c-b-a│
└   ┘		 ┌ ┐ ┌ ┐ ┌ ┐
│[1] 2 1 │ │ x│  │ 2│
│ 0 [2] -2 │ │ y│ = │ 6│
│ 0 0 [5]│ │ z│  │-10│
└ ┘ └ ┘ └ ┘
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The immediate question then is: for what values of ''a'', ''b'', and ''c'' is this system solvable?

----		 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.
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== Trivial Solvability == === Second Example ===
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The above system can be easily shown to be solvable for some values. Consider a system with '''free variables''':
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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.

----



== General Solvability ==

----



== 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'''.		 This system is rewritten as a linear system and eliminated into:
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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
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w + 2x + 2y + 2z = 1
w + 2(0) + 2y + 2(0) = 1
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The particular solution is: ''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:
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[-2 0 3/2 0]        ┌ ┐ ┌ ┐
       │ -2 │ │ 2 │
x = C │ 1 │ + C │ 0 │
 0 1│ 0 │ 2│ -1 │
       │ 0 │ │ 1 │
       └ ┘ └ ┘
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Next, compute the [[LinearAlgebra/NullSpaces|null space]].

Finally, the complete solution is the particular solution plus any combination of the null space.		 Altogether, the complete solution for the second example above is:
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The values ''C,,1,,'' and ''C,,2,,'' are any number, because any combination of the two vectors is valid for the complete solution.

General Solution

Elimination and other factorizations offer a method for obtaining the particular solutions (xp) of a linear system. The complete solution (xc) is the generalized answer.

Contents

  1. General Solution
    1. Introduction
      1. First Example
      2. Second Example
    2. Complete Solution

Introduction

First Example

The example used here 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: 

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

This system is rewritten as a linear system and eliminated into: 

┌           ┐ ┌  ┐   ┌  ┐
│[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 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 xc = xp + x0. That is, the 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 solutions. Again, for the second example above: 

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

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

┌    ┐     ┌    ┐     ┌    ┐
│ -2 │     │ -2 │     │  2 │
│  0 │ + C │  1 │ + C │  0 │
│ 3/2│    1│  0 │    2│ -1 │
│  0 │     │  0 │     │  1 │
└    ┘     └    ┘     └    ┘

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LinearAlgebra/GeneralSolution (last edited 2026-02-04 04:32:14 by DominicRicottone)