General Solution

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

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 + 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 │
     └    ┘     └    ┘     └    ┘

CategoryRicottone