Let $\ T: V\times W \rightarrow \mathbb V\otimes W$ be a map defined as $\ T(v,w) = v\otimes\ w$ where $\ v \in V,w\in W $. Then T is bilinear. Further if $\ (v_1,\ldots,v_n)\ and\;\ (w_1,\ldots,w_m)$ are bases of V and W respectively, then $(v_1\otimes w_1,\dots,v_n \otimes w_m)$ form a basis of $ V \otimes W$. I get the bilinearity part. But what does that kind of a basis say about the dimension of $V\otimes W$. Is the dimension $max(m,n)$, or am I missing something here?
Question regarding Tensor product
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0The notation can be confusing; an inner product is an example of a more general phenomenon of "pairing," where you take two elements of two vector spaces (or modules) and evaluate them against each other. If they're finite dimensional and you can prove this pairing is nondegenerate, it induces an isomorphism between the vector spaces. In fact, the inner product $\langle\cdot,\cdot\rangle$ may be regarded as a pairing between $V^*$ and $V$, which induces an isomorphism between them (when they're finite dimensional). – 2012-04-14
2 Answers
First, you're mixing up the universal property of the tensor product. Let $V,W$ be real vector spaces. Here's the correct statement: If $T:V\times W\to X$ is a bilinear map of real vector spaces, there exists a unique linear map $\tilde{T}:V\otimes W\to X$ which makes the following diagram commute: $\begin{matrix} V\times W & \xrightarrow{T} & X \\ \downarrow & \nearrow \tilde{T} & \\ V\otimes W & & \end{matrix}.$
Second, you're correct that bases $\{v_i\}$ of $V$ and $\{w_j\}$ of $W$ induce a basis $\{v_i\otimes w_j\}$ of $V\otimes W$. So to see the dimension of $V\otimes W$, just count! How many basis elements $\{v_i\otimes w_j\}_{i=1,\ldots,\dim V;j=1,\ldots,\dim W}$ are there? Answer: $(\dim V)(\dim W)$.
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0Thanks to everyone for those immediate answers. I just did not see the pairing involved in the basis which led me to the max(m,n) conclusion. – 2012-04-14
If $V$ is $n$ dimensional, and $W$ is $m$ dimensional, then $V \otimes W$ has dimension $nm$.
See http://mathworld.wolfram.com/VectorSpaceTensorProduct.html