Solutions to a system of Linear Equations

In summary, the number of distinct solutions of a system of linear equations over a field Z_p is either 0 or a power of p. To prove this, one can show that the set of solutions to the homogeneous equivalent of the system is a subspace of Z_p^n, and if the system is not homogeneous, any other solution can be represented as an element of the direct sum of a particular solution and the null space, which has a cardinality of a power of p. This relies on the understanding of the structure of solutions and the properties of Z_p as a field.
  • #1
Treadstone 71
275
0
"Show that the number of distinct solutions of a system of linear equations (any number of equations and unknowns) over a field Z_p is either 0, or a power of p."

I don't know where to start. Suppose there are n unknowns, if only I can show that the solution space is a subspace of Z_p ^n, then it's easy. But I can't seem to do it. Any hints?
 
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  • #2
what are your thoughts on the question? Have you any ideas on where to start?
 
  • #3
Do you know anything about groups?

I think that could help at looking at it.

Another question to ask yourself if you don't know the above is...

...what characteristics must p have for Z_p to be a field?
 
  • #4
Treadstone 71 said:
I don't know where to start. Suppose there are n unknowns, if only I can show that the solution space is a subspace of Z_p ^n, then it's easy. But I can't seem to do it. Any hints?

It's not always a subspace, but you should know something about the structure of the solutions. (if the Z_p is causing problems, what does it look like in the real case?)
 
  • #5
I've shown that the set of solutions to the HOMOGENEOUS equivalent of the system of equations is a subspace of Z_p ^n. I've also shown that if the system of equations is not homogeneous, and suppose that there is one solution v, then any other solution is an element of the direct sum of v + null T, which cardinality is a power of p because it's a subspace.
 

1. What is a system of linear equations?

A system of linear equations is a set of two or more equations with two or more unknown variables. The solution to a system of linear equations is the set of values that satisfy all of the equations in the system simultaneously.

2. How do you solve a system of linear equations?

There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. The most common method is Gaussian elimination, which involves using row operations to reduce the system to row echelon form and then back-substituting to find the solution.

3. Can a system of linear equations have more than one solution?

Yes, a system of linear equations can have infinitely many solutions, one unique solution, or no solutions. This depends on the number of equations and variables in the system and how they are related to each other.

4. How do you know if a system of linear equations has no solution?

If a system of linear equations has no solution, it means that the equations are inconsistent and cannot be satisfied simultaneously. This can be determined by looking at the coefficients of the equations and checking if they are consistent or contradictory.

5. Can a system of linear equations be solved using a calculator?

Yes, a system of linear equations can be solved using a calculator by using the matrix function. The equations can be written in matrix form and then solved using the appropriate function on the calculator. However, it is important to note that the calculator may not be able to show all the steps involved in the solution process.

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