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 :
- Two systems are equivalent.
- If $A$ and $B$ are matrices of coefficients of respective systems, $A$ and $B$ are row equivalent.
- $AX=0$ and $BX=0$ have same set of solutions.
- $A$ 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 :
- Two systems are equivalent.
- If $A'$ and $B'$ are respective augmented matrices, then $A'$ and $B'$ are row equivalent.
- $AX=Z_1$ and $BX=Z_2$ have same (non-empty)set of solutions.
- $A'$ and $B'$ have same row reduced echelon form.
Have I listed correctly. Thanks.
Best Answer
Yes, you are correct.
References below are to your text Linear Algebra.
For homogeneous systems,
Those items are overkill as a proof, but I included more than necessary for completeness.
For non-homogeneous systems,