Question

线性代数(英文) 求高手！！！！！

Let the transformation T : R2 → R2 be the reflection of any point through the origin.
(1) Prove that T is a linear transformation and find its matrix.
(2) Prove that there exist two lines L1 and L2 such that the composition of the reflections R1 and R2
about L1 and L2 has the same effect as T. What are the lines?
(3) Prove that T is in fact a rotation.
(4) Prove the generalization of (b)-(c) is also true: for any two lines L1 and L2 through the origin,the composition R1 ◦ R2 of the reflections R1 and R2 about L1 and L2 is a rotation.

求第二问和第四问的证明过程，只需要这两问！！！

Answer 1

(2)
T将每个点变换到与相对原点的对称点，因此x，y坐标将互换。
那么设L1为x轴，L2为y轴，那么任意点分别经L1、L2反设变换将与T等价。

(4)
设平面上一点P 及L1、L2 与 x轴正方向的夹角是 t, a, b。
首先，因为L1、L2经过原点，所以反射后的P离原点距离不变；
其次，反射后的夹角为：2b - (2a - t) = t + (2b+2a)。
综上，分别经L1、L2反射与将点旋转2b+2a角度等价。