# equivalent conditions to check if given two systems of linear equations are equivalent.

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.

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

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.