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Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.

Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)

Thanks

  • 7
    Have you read the following? http://math.stackexchange.com/questions/18881/motivation-for-tensor-product http://math.stackexchange.com/questions/76102/tensor-product-as-a-colimit http://math.stackexchange.com/questions/51155/understanding-of-the-tensor-product-of-vector-spaces http://math.stackexchange.com/questions/100633/what-is-the-categorical-diagram-for-the-tensor-product http://math.stackexchange.com/questions/17566/universality-of-tensor-product2012-03-02
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    @WillieWong : Thank you very much. Yes I have. However, the links above just mentioned the algebraic properties/structures. What I could not imagine is the concrete picture of tensor product.2012-03-02
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    The tensor product is an algebraic convenience and does not in general admit a geometric interpretation. (On the other hand, it is easy to give a geometric interpretation of the wedge product.)2012-03-02
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    Do you have a geometric appreciation for bilinear forms?2012-04-01
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    @eduardo: Related: http://math.stackexchange.com/questions/309838/the-physical-meaning-of-the-tensor-product#comment674782_3098382013-02-23

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