(a) Let $V$ be a complex vector space of dimension $n$.

What is a Hermitian form on $V$ ?

Given a Hermitian form, define the matrix $A$ of the form with respect to the basis $v_{1}, \ldots, v_{n}$ of $V$, and describe in terms of $A$ the value of the Hermitian form on two elements of $V$.

Now let $w_{1}, \ldots, w_{n}$ be another basis of $V$. Suppose $w_{i}=\sum_{j} p_{i j} v_{j}$, and let $P=\left(p_{i j}\right)$. Write down the matrix of the form with respect to this new basis in terms of $A$ and $P$.

Let $N=V^{\perp}$. Describe the dimension of $N$ in terms of the matrix $A$.

(b) Write down the matrix of the real quadratic form

$$x^{2}+y^{2}+2 z^{2}+2 x y+2 x z-2 y z .$$

Using the Gram-Schmidt algorithm, find a basis which diagonalises the form. What are its rank and signature?

(c) Let $V$ be a real vector space, and $\langle,$,$rangle a symmetric bilinear form on it. Let A$ be the matrix of this form in some basis.

Prove that the signature of $\langle,$,$rangle is the number of positive eigenvalues of A$ minus the number of negative eigenvalues.

Explain, using an example, why the eigenvalues themselves depend on the choice of a basis.