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I am reading Linear Algebra by Hoffman and Kunze. The definition for equivalent system is

Two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other system.

While going through one of the exercise, it was asked to check if given two systems are equivalent. I find it bit lengthy to check if each equation from each system can be written as linear combination of other equations from other system. So I look for some tricks and read few related questions asked here. After spending some time, I came to know about following conditions which were scattered here and there. I wanted to have them in one place and check if it's correct.

For two homogenous systems of equations, following statements are equivalent :

  1. Two systems are equivalent.
  2. If $A$ and $B$ are matrices of coefficients of respective systems, $A$ and $B$ are row equivalent.
  3. $AX=0$ and $BX=0$ have same set of solutions.
  4. $A$ and $B$ have same row reduced echelon form.

For two non-homogenous systems of equations, say $AX=Z_1$ and $BX=Z_2$, following statements are equivalent :

  1. Two systems are equivalent.
  2. If $A'$ and $B'$ are respective augmented matrices, then $A'$ and $B'$ are row equivalent.
  3. $AX=Z_1$ and $BX=Z_2$ have same (non-empty)set of solutions.
  4. $A'$ and $B'$ have same row reduced echelon form.

Have I listed correctly. Thanks.

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  • $\begingroup$ Are $X,Z_1,Z_2$ vectors? What does row equivalence mean? $\endgroup$
    – Sudix
    Commented Feb 22, 2021 at 12:41

1 Answer 1

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Yes, you are correct.

References below are to your text Linear Algebra.

For homogeneous systems,

  • 1 and 2 imply each other by the definitions of linear combination and equivalent on page 4, elementary row operations on page 6, and row-equivalent on page 7,
  • 1 implies 3 by Theorem 1.1 (Chapter 1, Theorem 1),
  • 2 and 4 imply each other by the first corollary to Theorem 2.11 and the fact that row-equivalence is an equivalence relation (page 7),
  • 2 implies 3 by Theorem 1.3, and
  • 3 implies 4 by the discussion after the proof of Theorem 1.5 and the fact that row-equivalence is an equivalence relation.

Those items are overkill as a proof, but I included more than necessary for completeness.

For non-homogeneous systems,

  • 1 and 2 imply each other by the definitions of linear combination, equivalent, elementary row operations, and row-equivalent,
  • 1 implies 3 by Theorem 1.1,
  • 2 and 4 imply each other by the first corollary to Theorem 2.11 and the fact that row-equivalence is an equivalence relation, and
  • 3 implies 4 by the discussion after Theorem 1.7 and the fact that row-equivalence is an equivalence relation.
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