Let and . Suppose that the system of linear equations Ax=b has a solution in . Does it necessarily have a solution in ?
and I thought I'd give an interesting, possibly wrong, approach to solving it. I'm not sure if such things can be done, if not maybe you can help me refine.
I considered the form of the equality as
where is a column vector of A. I then noticed that for then, and this is where I think I'm doing something forbidden, each x has the represenation
where is a distinct irrational number, , and p is the number of such distinct irrational numbers. I wound this out, but there may be a discrepancy with p and m. I feel this method can lead me to the answer, but I'm not sure where to go from here.
I end up getting something like this, I believe, after substitution:
Here, is the vector
I think there is no discrepancy with p and m because , and , so