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Fredrik

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## Main Question or Discussion Point

I'm reading the Wikipedia article, trying to understand the definition of the tensor product [itex]V\otimes W[/itex] of two vector spaces V and W. The first step is to take the cartesian product [itex]V\times W[/itex]. The next step is to define the "free vector space" [itex]F(V\times W)[/itex] as the set of all linear combinations of members of [itex]V\times W[/itex]. But how does that make sense when we haven't even defined the sum of two members of [itex]V\times W[/itex]?

I'm tempted to interpret the linear combination as just a string of text at this point, but then I can't make sense of the claim that the [itex]e_{v\times w}[/itex] are taken to be linearly independent for distinct [itex]v\times w[/itex].

Can someone help me make sense of this definition?

I'm tempted to interpret the linear combination as just a string of text at this point, but then I can't make sense of the claim that the [itex]e_{v\times w}[/itex] are taken to be linearly independent for distinct [itex]v\times w[/itex].

Can someone help me make sense of this definition?