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The tensor product of the linear spaces $M(m \times n)$ to $M(p \times m)$. Here, we let the elemantary tensors $A \otimes B$ correspond with the lineair transformation $X \to BXA^T$. The tensor product of $A$ and $B$ can be identified with the linear transformation $X\to BXA^T$ and the transformation matrix of this linear transformation that we obtain after vectorisation is is precisely the Kronecker product of $A$ and $B$. Prove this.

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Yeah, prove this. Good luck. –  Martin Brandenburg Feb 11 '13 at 19:29
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Did you write it down in index notation? –  tom Feb 11 '13 at 19:39
    
@MartinBrandenburg: for a new user, your comment is a bit cryptic. It sounds rude unless prefaced with a history of complaints about questions asked in the imperative voice. If you wish your comment to be effective, simply ask the author to not use the imperative voice or any of the other grammatical constructs that MSE finds rude or offensive. –  robjohn Feb 12 '13 at 0:16
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@user48288: since you mentioned "as it is stated in my book", and since you are new here, you may benefit from reading this guideline. –  Willie Wong Feb 12 '13 at 13:53
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As to index notation, tom's asking whether you considered expanding $(BXA^T)_{ij} = \sum_{k,l} B_{ik}X_{kl} A_{jl}$ and taking it from there. –  Willie Wong Feb 12 '13 at 13:54
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