Show that \(F^{m \times n}\) is isomorphic to \(F^{mn}\).
Solution
Define \[ \Phi:F^{m\times n}\longrightarrow F^{mn} \] by listing the entries row by row: \[ \Phi(A)=(A_{11},\ldots,A_{1n},A_{21},\ldots,A_{mn}). \] This map is linear because matrix addition and scalar multiplication are entrywise. It is injective because \(\Phi(A)=0\) forces every entry of \(A\) to vanish, and it is surjective because any \(mn\)-tuple can be placed, in order, into the entries of an \(m\times n\) matrix. Thus \(F^{m\times n}\cong F^{mn}\).