Is a Homogeneous System of Linear Equations Inconsistent When n Exceeds r?

[jdstokes]

True of false. A homogeneous system of $r$ linear equations in $n$ unknowns is inconsistent if the number of equations, $n$ exceeds the number of unknowns, $r$. The questions seems to be implying that $n=r$, in which case the system is consistent. Is this true?

Thanks.

James

 

[geosonel]

True of false. A homogeneous system of $r$ linear equations in $n$ unknowns is inconsistent if the number of equations, $n$ exceeds the number of unknowns, $r$. The questions seems to be implying that $n=r$, in which case the system is consistent. Is this true?

Thanks.

James

FALSE
any number of equations CAN BE consistent with any number of unknowns.

here are 5 consistent equations in 2 unknowns:

x + y = 2
2x + 2y = 4
3x + 3y = 6
4x + 4y = 8
5x + 5y = 10

or:

x + y = 0
2x + 2y = 0
3x + 3y = 0
4x + 4y = 0
5x + 5y = 0

etc.

 

[thepatientmental]

The statement is true. If the number of equations, n, exceeds the number of unknowns, r, then the system is inconsistent. This is because in a homogeneous system, all the equations must have the same number of unknowns. If there are more equations than unknowns, it means that at least one equation will have more unknowns than the others, making it impossible for the system to have a solution that satisfies all the equations. In other words, the system is inconsistent because there is no solution that can satisfy all the equations. Therefore, in order for a homogeneous system to be consistent, the number of equations must be equal to or less than the number of unknowns, n must be equal to or less than r.