A First Course in Linear Algebra · Problem C23

Systems of Linear Equations — Robert Beezer

Mathematics — Linear Algebra · solve-linear-system-solution-set

Problem

Find the solution set of the system of linear equations. Give the values of nn and rr, and interpret your answers in light of the theorems of this section. x1−2x2+x3−x4=3x1+x2+x3−x4=1x1+x2+x3−x4=2\begin{aligned} x_1 - 2x_2 + x_3 - x_4 &= 3\\ x_1 + x_2 + x_3 - x_4 &= 1\\ x_1 \phantom{{}+x_2} + x_3 - x_4 &= 2 \end{aligned}
Intuitive explanation

This problem asks you to find all solutions to a system by reducing it to row echelon form, then interpreting the rank rr and number of variables nn using the rank-nullity theorem. The key insight is that the number of free variables equals n−rn - r, which tells you whether the solution is unique, infinitely many, or empty—and the structure of the solution set (a point, line, plane, etc.) follows directly from that count.

Intuitive explanation

This problem asks you to find all solutions to a system by reducing it to row echelon form, then interpreting the rank rr and number of variables nn using the rank-nullity theorem. The key insight is that the number of free variables equals n−rn - r, which tells you whether the solution is unique, infinitely many, or empty—and the structure of the solution set (a point, line, plane, etc.) follows directly from that count.

Related theory

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