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Mathematics
Linear and Abstract Algebra
Prove that dim(V⊗W)=(dim V)(dim W)
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[QUOTE="member 587159, post: 6408541"] This is true because they define it that way. By definition, ##e_i \otimes f_j## is the bilinear map defined by $$e_i \otimes f_j: V^* \times W^* \to k: (\phi, \psi) \mapsto \psi(e_i) \psi(f_j)$$ It is just a definition. The only thing you should check is that this map is indeed ##k##-bilinear. We then obtain that ##e_i \otimes f_j \in V \otimes W## and it is then checked that ##\{e_i \otimes f_j\}_{(i,j)}## is a basis for ##V \otimes W## and since this basis has ##\dim V \dim W## amount of elements, you can conclude ##\dim(V \otimes W) = \dim V \dim W##. [/QUOTE]
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Forums
Mathematics
Linear and Abstract Algebra
Prove that dim(V⊗W)=(dim V)(dim W)
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