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Here are 4 versions of one theorem:

  1. Guillemin and Pollack:

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(Note that $A^\ast T(v_1,\dots,v_k)=T(Av_1,\dots,Av_k)$.)

  1. Lee:

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  1. Spivak:

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  1. Tu:

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I can see that Tu's version implies Spivak's version because the former works for all $v_1,\dots,v_n$ whereas the latter works for $v_1,\dots v_n$ a basis (note that Spivak's $(a_{ij})$ is the transpose of Tu's $A$, but this doesn't matter). Also, Lee's version implies Guillemin and Pollack's as well as Spivak's. (To see why it implies Spivak's version, let $(v_1,\dots,v_n)$ be a basis for $V$ and define $T$ by $T(v_i)=w_i$ for all $i$; then if $A$ is the matrix of $T$ w.r.t. this basis, then $A=(a_{ij})^t$ where $(a_{ij})$ is Spivak's matrix.)

That's all equivalences I can see so far. But they all seem so similar. Are they all equivalent? If not, is there a subset of the set of the above theorems the elements of which are pairwise equivalent?

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