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 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:

Altogether, the complete solution for the second example above is x_c = x_p + x₀.

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 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 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.

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