L1c. Generalized linear algebra: the pairing.

Instead of doing geometry, we were doing linear algebra. Let us now generalize this linear algebra before doing any geometry. This means looking at $V \oplus V^*$.

Notice that we have not added an arbitrary vector space to $V$, we have added its dual space $V^*$ and we cannot dismiss this. How do we keep track of this fact in a nice way? By defining a (canonical) pairing.

This is a good moment to introduce some conventions. We will normally use the capital letters $X,Y$ for elements of $V$, and the greek letters $\xi, \eta$ (xi and eta) for elements of $V^*$. For the elements of $V\oplus V^*$ we will use $v, w$. Thus, $v=X+\xi, w=Y+\eta$  are the usual elements of  $V\oplus V^*$.

And finally the pairing $\langle \; , \; \rangle$:
$$\langle X+\xi, Y+\eta \rangle = \frac{1}{2} (i_X \eta + i_Y \xi ).$$
Note that $\langle X+\xi, X+\xi \rangle = i_X \xi$.

This pairing has signature $(n,n)$. Given bases $\{ e_1,\ldots,e_n\}$ and $\{ e^1, \ldots, e^n \}$ of $V$  and $V^*$, the matrix of the pairing is written as
$$\left( \begin{array}{cc} 0 & \frac{1}{2}\textrm{Id} \\ \frac{1}{2}\textrm{Id} & 0 \end{array} \right)$$
for the basis $\{ e_1,\ldots,e_n, e^1, \ldots, e^n \}$ of $V \oplus V^*$, or as
$$\left( \begin{array}{cc} \textrm{Id} & 0 \\ 0 & -\textrm{Id} \end{array} \right)$$ for the basis  $\{ e_1+e^1,\ldots,e_n+e^n, e^1-e^1,\ldots,e_n-e^n \}$, from where we clearly see the signature.

Note that the pairing between any two elements of V, or any two elements of V^* is zero. This is what we call being isotropic.

Definition:  A vector subspace $L$ of a metric vector space $(W,\langle\;,\;\rangle )$ is said to be isotropic if $\langle x, y \rangle = 0$ for all $x,y \in L$.