Does an overdetermined system always have no solutions?

Question

What is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?

Answer 1

An overdetermined system (more equations than unknowns) is not necessarily a system with no solution. If one or more of the equations in the system (or one or more rows of its corresponding coefficient matrix) is/are (a) linear combination of the other equations, so the such a system might or might not be inconsistent.

And some systems which are not overdetermined (number of equations = number of unknowns) have no solutions.

What is true is that whenever we have an inconsistent system of equations, there is no solution.

My point is that you understand that a system of linear equations has no solution if and only if it is inconsistent.

Answer 2

Think of a plane. Every point has an $x$ and a $y$ component. A relationship between $x$ and $y$, say $2x = 3y+1$, is a property that some points have and others do not; $\langle 2,1\rangle$ has this property, and $\langle 0, 0\rangle$ does not. The set of points with this property forms some curve in the plane; typically a straight line.

Now suppose we have two equations, say $2x = 3y+1$ and $3x = 2y+4$; there are now two properties we want our points to possess. The points with one property lie on one line, and the points with the other property lie on another line; if the lines intersect, as they usually do, there will be exactly one point on both lines, and therefore one point that has both properties, and one pair of $x$ and $y$ that satisfies both equations.

If we have three properties, there will be three lines. The three lines will not usually intersect in the same place:

There will be a point with properties $A$ and $B$, a point with properties $A$ and $C$, and a point with properties $B$ and $C$. But no point will have all three properties because no point lies on all three lines.

Sometimes the three lines will all intersect in the same place. Then the three equations have a solution:

For example, $$\begin{align}2x&=3y+1\\ 3x &= 2y+4\\ x &= y+1 \end{align}$$

all intersect at the point $\langle 2,1\rangle$. So it can happen sometimes, but not always.