[Math] Number of solutions in system of linear equations

matricessystems of equations

I'm studying System of linear equations.

When solving Ax=b, it is said that the system can behave in 3 ways.

  • No solution
  • Unique solution
  • Infinitely many solutions

And according to Wikipedia page,

Usually, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system.

Usually, a system with the same number of equations and unknowns has a single unique solution.

Usually, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.

I think the word Usually implies that the system can behaves in 3 ways for any arbitrary matrices(it does not care the matrix is a square matrix or non-square matrix).

I want to know the possible behaviors when the matrix A is non-square.

My questions are

  1. Can non-square matrices(m < n) have a unique solution(unknowns > equations)?
  2. Can non-square matrices(m > n) have infinitely many solutions(unknowns < equations)?

Best Answer

  1. No. $m<n$ $\Rightarrow$ $\text{rank} A\le m<n$ $\Rightarrow$ $\dim\ker A=n-\text{rank} A\ge n-m>0$.
  2. Yes. For example, $0_{m\times n}x=0_{m\times 1}$, i.e. $m>n$ many trivial ($0=0$) equations.

In general,

  • no solution is always a possibility (e.g. one equation looks like $0=1$).
  • $\infty$ many solutions is always a possibility (e.g. one can add as many trivial equations $0=0$ as one wishes).
  • Unique solution is possible only for $m\ge n$.
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