Underdetermined Systems.
Relationship Between Unknowns and Equations
A linear system with more unknowns than equations (where m < n, with m as the number of equations and n as the number of unknowns) has either:
- Zero solutions, or
- Infinitely many solutions.
No Solutions
The "no solutions" condition occurs when the equations are logically inconsistent—no set of values for the unknowns can satisfy all equations simultaneously. For example:
- x = 1 and x = 2 cannot both be true, so the system has no solutions.
Infinite Solutions
The "infinite solutions" case is more nuanced. Equations act as constraints, limiting the possible values of the unknowns. Each equation reduces the flexibility of the system, but if there are fewer equations than unknowns, some freedom remains.
Equations as Constraints
An equation describes a relationship between variables, constraining their possible values. For example, 2x + 3y = 7 restricts x and y to pairs that satisfy the equation (e.g., (2, 1) or (0, 7/3)).
Adding more equations tightens these constraints, reducing the number of valid solutions. While "guess and check" becomes harder, even systematic methods require more work as constraints increase.
Dimension Reduction
"Each consistent equation reduces the solution space by one dimension." To understand this, consider a system with three unknowns (x, y, z) and no equations. The solution space is unconstrained—any point in 3D space is valid.
Adding one equation (e.g., x + y + z = 6) introduces a constraint. The solution space shrinks from 3D to a 2D plane, as two variables can be freely chosen, and the third is determined algebraically. For example:
- If x = 1 and y = 2, then z = 3.
- This leaves two "degrees of freedom" (choices for two variables).
The plane represents all valid solutions, extending infinitely in two dimensions but no longer in three.
Degrees of Freedom
With n unknowns and m consistent equations, the system has n - m degrees of freedom. For example:
- 3 unknowns + 1 equation = 2 degrees of freedom (a plane).
- 3 unknowns + 2 equations = 1 degree of freedom (a line).
- 3 unknowns + 3 equations = 0 degrees of freedom (a single point, if consistent).
If m < n, the system retains some freedom, allowing infinitely many solutions (unless the equations are inconsistent).