Characteristic property of the tensor product

Stub grade: A*

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Important linear algebra here. Totally

Statement

$\xymatrix{ V_1\times\cdots\times V_k \ar@2{->}[rr]^-A \ar@2{->}[d]_p & & W \\ V_1\otimes\cdots\otimes V_k \ar@{.>}[urr]_-{\overline{A} } }$
Diagram of the situation, the double-arrows is multilinear, the other is linear

Let $\mathbb{F}$ be a field and let $\big((V_i,\mathbb{F})\big)_{i\eq 1}^k$ be a family of finite dimensional vector spaces over $\mathbb{F}$. Let $(W,\mathbb{F})$ be another vector space over $\mathbb{F}$. Then:

• If $A:V_1\times\cdots\times V_k\rightarrow W$ is any multilinear map
  • there exists a unique linear map, $\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X$ such that:
    • $\overline{A}\circ p\eq A$ (that is: the diagram on the right commutes)

Where $p:V_1\times\cdots\times V_k\rightarrow V_1\otimes\cdots\otimes V_k$ by $p:(v_1,\ldots,v_k)\mapsto v_1\otimes\cdots\otimes v_k$ (and is $p$ is multilinear) (see claim 1 for the proof of this)

Proof of claims

Claim 1: $p$ is multilinear

• Let $i\in\{1,\ldots,k\}$ be given
  • $p(v_1,\ldots,v_{i-1},v_i+\lambda u,v_{i+1},\ldots,v_k)\eq v_1\otimes\cdots\otimes v_{i-1}\otimes (v_i+\lambda u)\otimes v_{i+1}\otimes\cdots\otimes v_k$

  $\eq (v_1\otimes\cdots\otimes v_k)+(v_1\otimes \cdots\otimes v_{i-1}\otimes \lambda u\otimes v_{i+1}\otimes\cdots\otimes v_k)$

  $\eq (v_1\otimes\cdots\otimes v_k)+\lambda(v_1\otimes\cdots\otimes v_{i-1}\otimes u\otimes v_{i+1}\otimes\cdots\otimes v_k)$

  • Next notice: $p(v_1,\ldots,v_k)+\lambda p(v_1,\ldots,v_{i-1},u,v_{i+1},\ldots,v_k)$

  $\eq (v_1\otimes\cdots\otimes v_k)+\lambda(v_1\otimes\cdots\otimes v_{i-1}\otimes u\otimes v_{i+1}\otimes \cdots \otimes v_k)$

  • We see: $p(v_1,\ldots,v_{i-1},v_i+\lambda u,v_{i+1},\ldots,v_k)$$\eq p(v_1,\ldots,v_k)+\lambda p(v_1,\ldots,v_{i-1},u,v_{i+1},\ldots,v_k)$
• Since $i\in \{1,\ldots,k\}$ was arbitrary, we see it is linear in each $i$ - the definition of a multilinear map

Proof

$\xymatrix{ \mathcal{F}(V_1\times\cdots\times V_k) \ar@{-->}@/^1.8em/[rrr]^{\tilde{A} } \ar[dr]_\pi & V_1\times\cdots\times V_k \ar@{_{(}->}[l]_-i \ar@2{->}[d]_p \ar@2{->}[rr]^-A & & W \\ & V_1\otimes\cdots\otimes V_k \ar@{.>}[urr]_-{\overline{A} } }$
Diagram

The gist of the proof is as follows:

• First note that by the characteristic property of the free vector space that $\overline{A}$ extends uniquely to a linear map $\tilde{A}:\mathcal{F}(V_1\times\cdots\times V_2)\rightarrow W$
• Now we want to factor $\overline{A}$ through $\pi$, this will yield us another linear map:

  $$\overline{A}:\underbrace{\dfrac{\mathcal{F}(V_1\times\cdots\times V_K)}{\mathcal{R} } }_{\text{i.e. }V_1\otimes\cdots\otimes V_k} \rightarrow W$$ - as required

  • To do this we use the group factorisation theorem:
    • If $\mathcal{R}\eq\text{Ker}(\pi)\subseteq\text{Ker}(\tilde{A})$ then $\tilde{A}$ factors through $\pi$ to yield (in this case) $\overline{A}$, and furthermore $\overline{A}$ is a linear map.
      • Let $(v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)\in\mathcal{R}$ be given
        • Then $\tilde{A}(v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)$ is:
          • $\eq A(v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)$ as it extends $A$ and $(v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)\in V_1\times\cdots\times V_k$
          • $\eq aA(v_1,\ldots,v_k)$ as $A$ is multilinear
          • $\eq a\tilde{A}(v_1,\ldots,v_k)$ as $\tilde{A}$ is an extension of $A$
          • $\eq \tilde{A}(a(v_1,\ldots,v_k))$ as $\tilde{A}$ is linear

Grade: A

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The gist is above. Tidy up a bit, basically done, not quite sure where to go, demote to E once the outline is complete