Differences between revisions 11 and 12
⇤ ← Revision 11 as of 2026-02-04 03:08:07 →
Size: 1857
Editor: DominicRicottone
Comment: Typo
← Revision 12 as of 2026-02-04 04:32:14 → ⇥
Size: 2186
Editor: DominicRicottone
Comment: Adding row space solution
Deletions are marked like this. Additions are marked like this.
Line 13: Line 13:
A consistent linear system has either one or infinitely many solutions. The general solution describes all of them. A consistent linear system has either one or infinitely many solutions. The '''general solution''' describes all of them.
Line 15: Line 15:
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.		 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.
Line 35: Line 33:
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: 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:
Line 37: Line 35:
{{{
       ┌ ┐ ┌ ┐
       │ -2 │ │ 2 │
x = C │ 1 │ + C │ 0 │
 0 1│ 0 │ 2│ -2 │
       │ 0 │ │ 1 │
       └ ┘ └ ┘
}}}		 {{attachment:null.svg}}
Line 46: Line 37:
Altogether, the complete solution for the second example above is: Altogether, the complete solution for the second example above is ''x,,c,, = x,,p,, + x,,0,,''.
Line 48: Line 39:
{{{
x = x + x
 c p 0		 {{attachment:complete.svg}}
Line 52: Line 41:
     ┌ ┐ ┌ ┐ ┌ ┐
     │ -2 │ │ -2 │ │ 2 │
x = │ 0 │ + C │ 1 │ + C │ 0 │
 c │ 3/2│ 1│ 0 │ 2│ -2 │
     │ 0 │ │ 0 │ │ 1 │
     └ ┘ └ ┘ └ ┘
}}}		 ----



== 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,,''.

General Solution

A complete solution is a generalization of a particular solution. It is generally notated as xc.

Contents

  1. General Solution
    1. Description
    2. Solution
    3. Row Space Solution

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:

null.svg

Altogether, the complete solution for the second example above is xc = xp + x0.

complete.svg

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.

CategoryRicottone

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