Book about Tensor Product of Vector Spaces

Question

This subject is very commun for any book about modules. However, some undergraduate majors have Linear Algebra course before the Abstract Algebra one (where we treat about modules) and it is not common to find Linear algebra textbooks dealing with tensor product of vector spaces. What is a good reference with exercises, good theory development, properties and so on tensor products?

Answer 1

For vectors $a,b$, the notation $a\otimes b$ is basically stand to mean not much more or less than just the pair $(a,b)$. For vector spaces $A,B$, $\ A\otimes B$ is the set of all 'formal' sums of $a\otimes b$'s, and we require the following equalities: $$ \begin{aligned} (a_1+a_2)\otimes b&=a_1\otimes b \ + \ a_2\otimes b \\ a\otimes (b_1+b_2) &=a\otimes b_1 \ + \ a\otimes b _2 \\ (\lambda\cdot a)\otimes b &= a\otimes(\lambda\cdot b) \end{aligned}$$ The important feature of tensor product is the following:

Thm. The bilinear functions $A\times B\to V$ are in a one-to-one correspondence with the linear functions $A\otimes B\ \to V$.

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To understand tensor product, I would suggest to start with less structure. If we forget about addition, in place of vector spaces or modules we arrive to monoid actions: a monoid is a set equipped with an associative binary operation and a unit element for that operation, and a right action of a monoid $M$ on set $S$ is a function $S\times M\to S$ (writing $sm$ for the image of $(s,m)$) such that $$s1=s \quad \quad (sa)b=s(ab)$$ for all $s\in S$ and $a,b\in M$. A left action is defined dually.

Now, if we have a right action $S\times M\to S$ and a left action $M\times T\to T$, then their tensor product (which will be a set) is defined as $$S\otimes_M T:=(S\times T) \ / \sim $$ where $\sim$ is the equivalence relation generated by $(sm,t)\sim (s,mt)$.

In other words, $S\otimes_M T$ consists of pairs $(s,t)\in S\times T$, and two pairs are considered equal whenever one can be written as $(sm,t)$ and the other as $(s,mt)$ using the same elements $s\in S,\ t\in T,\ m\in M$.

Most of the main features of tensor product of vector spaces is already true for monoid actions. E.g., if $M$ acts from the right on set $B$, then the set $\hom(B,C)$ of functions from $B$ to another set $C$ can be equipped with a left action of $M$: $$m\cdot f := \ b\mapsto f(bm)\,. $$ Moreover, we then have $\ \hom_M(A,\hom(B,C)) \cong \hom(A\otimes_M B,C) $, if $M$ acts on $A$ from left. (Here $\hom_M(X,Y)$ denotes the set of those functions $f:X\to Y$ which preserve the left action of $M$: $\ f(mx)=m\,f(x)$.)

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Now, if $M$ is happened to be commutative, then any left action on a set $S$ already implies a right action on $S$, setting $sm:=ms$, as then we have the necessary $(sa)b=s(ab)$, moreover it is a biact ${}_MS_M$ at once.