> I've never been in a Linear algebra, how does one calculate the dual of a vector space? And what is it used for?

Once you've chosen a basis, the dual of a vector is its transpose \$$v^T\$$. This is because \$$f(x)=v^T x\$$ is a linear functional on a the vector space.

> Here's a nice fact about the dual: the set of all linear maps from a vector space \$$V\$$ to a vector space \$$W\$$ is itself a vector space, and this vector space is isomorphic to \$$W \otimes V^\ast\$$.

While I am a little fuzzy on what the tensor product means for vectors spaces and what cap and cup should be here.

My intuition is that cap sends elements of \$$k\$$ to a diagonal matrix (which is an element of \$$V \otimes V^\ast\$$).

$\cap_V : a \mapsto a I \quad a \in k \quad I : V \to V$

While cup is something like the trace, sending linear maps in \$$V^\ast \otimes V\$$ to an element of \$$k\$$.

$\cup_V : A \mapsto \mathrm{tr}(A) \quad A : V^\ast \to V^\ast$

If you compose them directly, you get a linear map \$$k \to k\$$ (which is a field homomorphism I believe).

Is that right?