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I understand the particular form of vectors of subspace $S$ of Free vector space $F(V \times W)$;

$$ (v+v',w)-(v,w)-(v',w) $$

$$ (v,w+w')-(v,w)-(v,w') $$

$$ a(v,w)-(av,w) $$

$$ a(v,w)-(v,aw) $$

They are just the "right candidates" to force bilinearity. But, it seems that they are really a set of four vectors. I mean, the claim is that they generate the subspace $S$. Hence, we can write a linear combination with then.

So, can I say that every vector of the subspace $S$ is written like: $$u = c^{1}(v+v',w)-(v,w)-(v',w) + c^{2}(v,w+w')-(v,w)-(v,w')+c^{3}a(v,w)-(av,w)+c^{4}a(v,w)-(v,aw) \equiv c^{1}e_1 + c^{2}e_2 + c^{3}e_3 + c^{4}e_4 ?$$

Put in another way, they are a basis of four vectors like: $$\{e_1,e_2,e_3,e_4\} \equiv \{(v+v',w)-(v,w)-(v',w) , (v,w+w')-(v,w)-(v,w') , a(v,w)-(av,w) , a(v,w)-(v,aw)\}?$$

BasicMathGuy
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  • The subspace $S$ is generated by all elements of $F(V\times W)$ of the given form, where $v,v’,w,w’$ are arbitrary vectors in $V$ and $W$, and $a$ is an arbitrary scalar. – Arturo Magidin Apr 18 '20 at 03:09
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    So, no; there are more forms for vectors in $S$; $v$, $w$, $v’$, $w’$ are not fixed. For example, if $V=\mathbb{R}^2$ and $W=\mathbb{R}^2$, one generating vector for $S$ is $(e_1+e_2,e_1)-(e_1,e_1) - (e_2,e_1)$, and a different generating vector is $(e_1+2e_1,e_1) - (e_1,e_1) - (2e_1,e_1)$. If $V$ and $W$ are infinite (either infinite dimensional, or the field is infinite), then the set of vectors you are describing is infinite. – Arturo Magidin Apr 18 '20 at 03:11
  • Also: this – Arturo Magidin Apr 18 '20 at 03:13
  • I encourage you to write a answer, even though your comment was sufficient. – BasicMathGuy Apr 18 '20 at 03:15
  • @ArturoMagidin about your answer here https://math.stackexchange.com/questions/61916/understanding-the-details-of-the-construction-of-the-tensor-product

    Then for the element $(3,8)$ ou have a basis, for element $(10,4)$ you have another, and so on?

     – BasicMathGuy Apr 18 '20 at 03:20
  • Yes. And another for $(6,16)$, and so on. – Arturo Magidin Apr 18 '20 at 04:10
  • I think I already wrote an answer... – Arturo Magidin Apr 18 '20 at 04:11

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