Solvability of linear systems over finite fields
by John from John D. Cook on (#6PD9F)
If you haven equations in n unknowns over a finite field with q elements, how likely is it that the system of equations has a solution?
The number of possible n * n matrices with entries from a field of size q is qn^2. The set of invertible n * n matrices over a field with q elements is GLn(q) and the number of elements in this set is [1]
The probability that an n * n matrix is invertible is then
which is an increasing function of q and a decreasing function of n. More on this function in the next post.
Related posts- Spaces and subspaces over finite fields
- Finite projective planes
- Finite field Diffie Hellman encryption
[1] Robert A. Wilson. The Finite Simple Groups. Springer 2009
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