Approximate Solution

Explore the idea of an approximate solution for an inconsistent linear system.

We concluded our previous discussion about inconsistent linear systems by saying that no solution exists for such a system. However, there are scenarios in which it’s still important to find the solution for a system, even if it’s an approximate solution. Imagine a space travel company is landing its rocket booster back on earth. The company’s goal is to land the rocket on a platform floating on the sea. Assume that the mathematical model for this task is a linear system with variables like wind speed, rocket weight, and sea waves. The ideal case would be to land the rocket on the marker at the center of the platform.

However, let’s assume that the linear system turns out to be an inconsistent system. Now, instead of aborting the landing because an exact solution doesn’t exist to land the rocket on the marked point, it’s in the company’s best interest to land the rocket anywhere on the platform. This may be achieved by calculating an approximate solution for the linear system.

Approximation

In contrast to the exact solution that satisfies all equations of the linear system, an approximate solution is a point that nearly satisfies all the equations.

A point satisfies an equation if, by replacing the variables of the equation with the values of the point, the equality is maintained.

Exact Solution Near Satisfaction
a1x1+a2x2++anxn=ba_1x_1+a_2x_2+\cdots+a_nx_n = b a1x1+a2x2++anxnb=0a_1x_1+a_2x_2+\cdots+a_nx_n -b = 0 a1x^1+a2x^2++anx^nba_1\hat x_1+a_2\hat x_2+\cdots+a_n\hat x_n \approx b
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