A First Course in Linear Algebra · Problem T11
Systems of Linear Equations — Robert Beezer
Mathematics — Linear Algebra · row-equivalence-equivalence-relation
Problem
Suppose that , and are matrices. Use the definition of row-equivalence (Definition REM) to prove the following three facts. 1. is row-equivalent to . 2. If is row-equivalent to , then is row-equivalent to . 3. If is row-equivalent to , and is row-equivalent to , then is row-equivalent to . A relationship that satisfies these three properties is known as an equivalence relation, an important idea in the study of various algebras. This is a formal way of saying that a relationship behaves like equality, without requiring the relationship to be as strict as equality itself. We will see it again in Theorem SER.Intuitive explanation
Row-equivalence means you can transform one matrix into another through a sequence of elementary row operations—and the key insight is that these operations are reversible and composable. To prove these three properties, you'll need to show that: (1) doing nothing is a valid sequence, (2) reversing any sequence of operations gives you back the original, and (3) chaining two sequences together works. This demonstrates that row-equivalence is reflexive, symmetric, and transitive—the hallmarks of an equivalence relation.
Intuitive explanation
Row-equivalence means you can transform one matrix into another through a sequence of elementary row operations—and the key insight is that these operations are reversible and composable. To prove these three properties, you'll need to show that: (1) doing nothing is a valid sequence, (2) reversing any sequence of operations gives you back the original, and (3) chaining two sequences together works. This demonstrates that row-equivalence is reflexive, symmetric, and transitive—the hallmarks of an equivalence relation.