Question

What would you have to know about the solution set of a homogenous system of 18 linear equations 20 variables in order to understand that every associated nonhomogenous equation has a solution? Discuss.

Short answer

The system of the equation has a solution for every b in \({\mathbb{R}^{18}}\).

Step-by-step solution

Make assumptions for the given problem

Let the system \(A{\bf{x}} = {\bf{b}}\) has 18 equations and 20 variables. Then, by the rank theorem, you get

\({\rm{rank}}\,\,A + \dim \,\,{\rm{Nul}}\,\,A = n\).

Find the solution set using rank theorem

Since the number of columns in A is 20, the column space of A will be

\(\begin{array}{c}\dim \,{\rm{Col}}\,\,A \ge 20 - 2\\ \ge 18\end{array}\).

Therefore, the column space of A is \({\mathbb{R}^{18}}\).

Hence, the system of the equation has a solution for every b in \({\mathbb{R}^{18}}\).