3 Systems of Linear Equations

Solving simultaneous equations is the oldest problem in the subject, and this section settles it completely.

The method is elimination, carried out on the augmented matrix by the row operations of Section 2. What the section adds is the theory that makes the method trustworthy: that row operations never change the solution set, so the simplified system may be solved in place of the original; and that the reduced echelon form answers, by inspection of a single column, whether a solution exists at all.

The counting is worth stating in advance. The variables split into leading and free; a system is inconsistent exactly when a row reads \(0=1\); and when it is consistent, the number of free variables is the number of independent choices in the answer. A homogeneous system with more unknowns than equations therefore always has a non-trivial solution — a fact which returns in Section 6 as the rank–nullity theorem.

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