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Characterization of rank two subspaces of a tensor product space Iwata, George Fumimaro

Abstract

Let U, V be two vector spaces of dimensions n and m, respectively, over an algebraically closed field F; let U⊗V be their tensor product; and let Rk(U⊗V) be the set of all rank k tensors in U⊗V, that is Rk(U⊗V) = {[formula omitted] are each linearly independent in U and V respectively}. We first obtain conditions on two vectors X and Y that they be members of a subspace H contained in Rk(U⊗V). In chapter 2, we restrict our consideration to the rank 2 case, and derive a characterization of subspaces contained in R2(U⊗V). We show that any such subspace must be one of three types, and we find the maximum dimension of each type. We also find the dimension of the intersection of two subspaces of different types. Finally, we show that any maximal subspace has a dimension which depends only on its type.

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