Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude that all of them would vanish.
If such a result is true, how many minors will do the job and which ones ?
I am wondering if it is even possible to calculate the value of all minors based on the value of a nicely chosen "generating subset" ?
Edit:- The question which I had asked does not have an affirmative answer as explained
by Steven Sam. But matrix minors do satisfy some relationships see the answer by Sheikraisinrollbank below. If someone can modify the question to a more appropriate one (in light of Steven Sam and Sheikraisinrollbank answers ) please feel free to do so.
I have often come across a situation (more so at present than ever before) where in order to answer a problem in my subject area I am led to questions which are totally different areas about which I have absolutely no familiarity. Most often these are quite basic and I would suppose well known to any one who works in those areas. It is natural that a person who is not familiar with a given field will end up asking for "a result of the following kind" rather than a precise question. For a person who is knowledgeable I understand the question may be irritating or look ill posed but mind you the hapless fellow is not a graduate student in the given field and please do not judge him accordingly. I think its desirable that if someone knows how to reformulate the question to something so that it becomes well posed or meaningful it should be done. Why not edit the question to something so that it becomes a valid well posed question, to something which is obviously much more interesting than which was originally posed ?
Best Answer
To counterbalance Steven Sam's answer some (b/c the OP's intuition is correct in a sense):
It's true that the right way to check that all m by m minors are zero in practice is Gaussian elimination. However, while the minors may be linearly independent, they satisfy quadratic relations ("Plucker relations", see for instance the wikipedia article on Grassmannians) that allow you to deduce some things. In the simplest non-trivial case of 2 by 4 matrices, writing $m_{ij}$ for the $(i,j)$th $2$ by $2$ minor one has $$m_{12}m_{34}-m_{13}m_{24}+m_{14}m_{23}=0.$$ This might have some theoretical value for the OP's situation that Gaussian elimination does not. For instance, in this case it allows one to deduce that if $m_{12}$ and $m_{13}$ are both zero then either $m_{14}$ or $m_{23}$ is zero. But it's hard to know if this helps without knowing somewhat more about the motivating problem.