I am self studying linear Algebra from Hoffman and Kunze and Couldn't think about how a deduction mist be true.
On page 15 authors wrote
Why if a solution $x_{1}$ ,..., $x_{n}$ belongs to F then the system of equations must have a solution with $x_{1} $,...,$x_{n}$ in $F_{1} $ ?
Is it is due to the fact that in AX=Y, both A and Y are belonging to $ F_{1}$ .so solution X must exist in $ F_{1} $ ?
Because the augmented matrix $[A|Y]$ has the same row-reduced echelon form in $F_1$ and $F$.