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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

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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. $