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| 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: | Furthermore, it was noted [[LinearAlgebra/NullSpace#Solutions|here]] that the null space vectors are ''[-2 1 0 0]'' and ''[2 0 -2 1]''. The null space solution is ''any'' linear combination of these vectors. Consider: |
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| 0 1│ 0 │ 2│ -1 │ | 0 1│ 0 │ 2│ -2 │ |
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| c │ 3/2│ 1│ 0 │ 2│ -1 │ | c │ 3/2│ 1│ 0 │ 2│ -2 │ |
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 -2 1]. The null space solution is any linear combination of these vectors. Consider:
┌ ┐ ┌ ┐
│ -2 │ │ 2 │
x = C │ 1 │ + C │ 0 │
0 1│ 0 │ 2│ -2 │
│ 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│ -2 │
│ 0 │ │ 0 │ │ 1 │
└ ┘ └ ┘ └ ┘