这是一份uwa西澳大学MATH3033的成功案例
几何学|MATH3033 Geometry代写 UWA代写
Concretely parametrize the sphere in the usual way:
$$
x_{1}=\sin \theta \sin \phi, \quad x_{2}=\sin \theta \cos \phi, \quad x_{3}=\cos \theta
$$
then with the poles removed the range of values is $0<\theta<\pi, 0 \leq \phi<2 \pi$. The antipodal map is
$$
\theta \mapsto \pi-\theta, \quad \phi \mapsto \phi+\pi .
$$
We can therefore identify the space of lines in $\mathbf{R}^{2}$ as the pairs
$$
(\theta, \phi) \in(0, \pi) \times[0, \pi]
$$
MATH3033 COURSE NOTES :
We can add symmetric bilinear forms: $(B+C)(v, w)=B(v, w)+C(v, w)$ and multiply by a scalar $(\lambda B)(v, w)=\lambda B(v, w)$ so they form a vector space isomorphic to the space of symmetric $n \times n$ matrices which has dimension $n(n+1) / 2$. If we take a different basis
$$
w_{i}=\sum_{j} P_{j i} v_{j}
$$
then
$$
B\left(w_{i}, w_{j}\right)=B\left(\sum_{k} P_{k i} v_{k}, \sum_{\ell} P_{\ell j} v_{\ell}\right)=\sum_{k, \ell} P_{k i} B\left(v_{k}, v_{\ell}\right) P_{\ell j}
$$
so that the matrix $\beta_{i j}=B\left(v_{i}, v_{j}\right)$ changes under a change of basis to
$$
\beta^{\prime}=P^{T} \beta P .
$$