I am using a result in a problem but I am not sure how to write or support it correctly.
We have an integer matrix $A$ over the reals.
We can define a matrix $A'$ over $\mathbb F_p$ by just taking each entry $\bmod p$.
I want to say that the rank of $A'$ is less than or equal to the rank of $A$.
How do I justify this?
I am guessing the proof first tells us if there is a linear combination for $0$ in $\mathbb R$ for the rows of $A$ then that linear combination must also exist in $\mathbb Q$, and then we can probably multiply everything by the lcm of the denominators so our coefficients are in $\mathbb Z$.
So I guess what we can do is show that if a set of integer vectors not independant over $\mathbb R$ then they are not independent when reduced $\bmod p$ in $\mathbb F_p$. I think that constitutes a proof.
Does anyone have a reference of this result or thoughts on the proof? (I'm guessing something similar may also exist talking about some algebraic setting instead of specific fields and rings).
Many thanks and best regards.