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I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure.

But just in the case of taking the tensor product of two vector spaces to get another vector space, how is this different than just taking the direct product of the two spaces? Because the direct product will also give us a larger, new vector space.

I'm guessing the answer to my question will be along the lines of "it's because of the Universal Property the tensor product satisfies..." but I can afford to be set a little straight here!

Thanks

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# Difference between tensor product and direct product?

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