> 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?

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?