Basis for space of all $m$ by $n$ matrices Question:  Let $(X_1,...,X_m)$ basis for $R^m$ and $(Y_1,...,Y_n)$ basis for $R^n$.  Is true that the $mn$ matrices $X_iY_j^t$ forms a basis for space of all $m$ by $n$ matrices?
I verified this for standard ordered basis.  But I have no idea how to proceed in general.  Can some one give suggestion to this problem?
Thanks

Take invertible matrices $P,Q$ such that $Pe_i = X_i$ and $\{e_i\}$ is the standard basis  for $\mathbb R^m$ and $Q f_j = Y_j$  and $\{f_j\}$  is the standard basis  for $\mathbb R^n.$ Given your target rectangular matrix $M,$ write
$$  \sum a_{ij} e_i f_j^T = P^{-1} M {Q^{-1}}^T.  $$
Then
$$   \sum a_{ij} X_i Y_j^T = M. $$
