Over-determined and Under-determined systems
How do I show that a system is both over-determined and under-determined? I am supposed to come up with a matrix that satisfies both but am not really sure I understand what types of equations would satisfy these criteria. If anyone could give me an example and maybe a format to go by, I would appreciate it.
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$\begingroup$A system of linear equations is called over-determined if there are more equations than unknowns. For example:
$$ \begin{cases} x+y = 1\\2x+y = 1\\ 3x+y = 1\end{cases}$$
Often times for an over-determined system, as in the example above, there will be no solution. That is not to say however that every over-determined system has no solution however. There might be exactly one or infinitely many solutions.
For example, $\begin{cases} x+y=1\\ 2x+y = 1 \\ 3x+2y=2\end{cases}$ has one solution ($x=0,y=1$) and $\begin{cases}x+y=1\\ 2x+2y=2 \\ 3x+3y=3\end{cases}$ has infinitely many solutions (all of the form $x=1-y$).
A system of linear equations is called under-determined if there are more unknowns than equations. For example:
$$\begin{cases} x+y+z = 1\\ x+2y+z = 1\end{cases}$$
Often times for an under-determined system, as in the example above, there will be infinitely many solutions. That is not to say however that every under-determined system has infinitely many solutions however. It is possible that there are no solutions.
For example, $\begin{cases} x+y+z = 1\\ x+y+z = 2\end{cases}$ has no solution (else $2=1$ which is untrue). Note that for the under-determined systems, it is impossible to have only one unique solution. It will always be either infinitely many or none.
The way the terms are defined has the number of equations vs unknowns to be strictly different. As such you will fall into only one case or the other and never both at the same time.
$\endgroup$ 7 $\begingroup$I believe that, as pointed out in Overdetermined and underdetermined systems of equation put simply, thinking of the equations in a system making up a set of requests (equations) to a certain number of people (unknowns) is helpful to understand why systems can be overdetermined or underdetermined in the first place.
In your specific case, imagine to have 3 people, all waiting for a command from you. Finally, you speak and say:
- Bob, go fetch a bottle of water
- Bob, help me paint the house
- Lisa, do your homework
This is a real world example of a system that is both under and overdetermined. What makes it underdetermined is the fact that you had 3 people in front of you, and only addressed 2 of them, whereas giving conflicting commands to Bob makes it an overdetermined one.
$\endgroup$ $\begingroup$An over-determined system is a system of equations that has more equations than unknowns (e.g. three equations and two unknowns).
An under-determined system is a system of equations that has more unknowns than vectors (e.g. three unknowns and two equations).
The combination of the two leads to the critical case which is when you have the same number of equations as you do unknowns. The best case scenario of this would be to start in reduced row echelon form and work your way backwards to come up with an example of such a matrix.
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