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The relevance in a sampling theory of U-invariant subspaces of a Hilbert space H, where U denotes a unitary operator on H, is nowadays a recognized fact. Indeed, shift-invariant subspaces of L-2(R) become a particular example; periodic extensions of finite signals also provide a remarkable example. As a consequence, the availability of an abstract U-sampling theory becomes a useful tool to handle these problems. In this paper, we derive a sampling theory for finite dimensional multiple generated U-invariant subspaces of a Hilbert space H. As the involved samples are identified as frame coefficients in a suitable euclidean space, the relevant mathematical technique is that of the finite frame theory. Since finite frames are nothing but spanning sets of vectors, the used technique naturally meets matrix analysis.