can we spcify a condition so that an overdetermined system have exactly one solution?

Question

caozeyuan · Answer

oh wait, I think maybe the answer is that some equations have to be dependent on others and when we get rid of those euqations from the ssytem, it comes out to be a consistant system with only one solution

caozeyuan · Answer

but how would we write this is precise mathematical terms?

caozeyuan · Answer

Ok, so if the system is consistant with only one solution, the corresponding coeffcient matrix have to be full rank, and so if we put the original matrix in RREF and ignore all 0 rows, we have an identity matrix

caozeyuan · Answer

is my reasoning correct so far?

holsteremission · Answer

You're correct that some of the equations/rows of the coefficient matrix will have to be dependent. If $\mathbf A$ has $m$ rows and $n$ columns, with $m>n$, then the only way for the system to remain consistent is if $m-n$ of the rows of $\mathbf A$ are linear combinations of other rows.

However, if $\mathbf A$ is rectangular then it can't possibly be of full rank. With $m>n$ it can have at most rank $n$ ("full column rank").