Row reduced echelon form and its meaning

[CStudent]

Hey.

I have the following question to solve:

* Given a matrix A that is size m x n and m>n.
Let R be the RREF that we get by Gaussian elimination of A.
Prove that the system equation Ax=0 has only one solution iff in every column of R there is a leading element.

I have some answer of intuition so I'm not really sure,
Let's assume that we had R with some free variable, and we know(?) that any free variable has a degree of freedom which means that it yields infinite number of solutions.

Now, I am not sure again about the establishment of this proof and to what extent it's accurate. Moreover, I am not if it proves the point of iff (equivalence).

Another similar question, but I have no idea what it means:

* Given a matrix A that is size m x n and m>n.
Let R be the RREF that we get by gaussian elimination of A.
Prove that for every

the system equation Ax=b has a solution iff R doesn't have rows of zeros.

Thank you!

 

[jvicens]

Hello,

Thank you for your question. I am happy to provide some insight and clarification for you.

Firstly, to prove that the system equation Ax=0 has only one solution, we need to show that the system is consistent (has a solution) and that the solution is unique (there are no other solutions).

In order for the system to be consistent, it must have a solution for every row of the matrix R. This means that every column of R must have a leading element (a non-zero element that is the first non-zero element in its column). This is because if there is no leading element in a column, it means that the corresponding variable is a free variable and can take on any value, leading to an infinite number of solutions.

On the other hand, if there is a leading element in every column of R, it means that every variable has a specific value and there is no degree of freedom, leading to only one unique solution for the system.

Therefore, the statement "iff in every column of R there is a leading element" is accurate and proves the point of equivalence.

For the second question, the statement "iff R doesn't have rows of zeros" means that every row in R must have at least one non-zero element. This is because if there is a row of zeros, it means that the corresponding equation is inconsistent and has no solution.

Therefore, for the system equation Ax=b to have a solution, R must not have any rows of zeros. This is because if there is a row of zeros, it means that the corresponding equation is inconsistent and has no solution. Conversely, if R does not have any rows of zeros, it means that every equation in the system is consistent and has a solution.

I hope this helps clarify the proof for you. Let me know if you have any further questions or require additional clarification.