I understood that two systems of linear equations are equivalent if one can be obtained...

Gauge Terrell

Gauge Terrell

Answered

2022-07-01

I understood that two systems of linear equations are equivalent if one can be obtained by the linear combination of the other system and vice versa. But can those two systems of equations be equivalent even if the solution x i = 0 ? ( 1 i n ).Example set:
x 1 x 2 = 0 ;
2 x 1 + x 2 = 0
and
3 x 1 + x 2 = 0 ;
x 1 + x 2 = 0
Are these systems of equations equivalent?

Answer & Explanation

Janiyah Patton

Janiyah Patton

Expert

2022-07-02Added 12 answers

It depends on your definition of equivalence. Are two systems of equations defined to be "equivalent" if their solution sets are equivalent as sets? Or are they ONLY equivalent if one system can be obtained by a linear combination of the other system (and vice versa)?
Brock Byrd

Brock Byrd

Expert

2022-07-03Added 2 answers

Note that the latter equivalence of systems by linear combinations naturally implies the former equivalence of their solution sets, but having equivalent solution sets does NOT imply that one system can be obtained by a linear combination of the other system.

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