矩阵 $\mathbf{M}$ 为

(a) 矩阵 $\mathbf{M}$ 表示在 $x$-$y$ 平面上依次进行的两个几何变换。

完整描述每个变换，并明确它们的施行顺序。

(b) 求矩阵 $\mathbf{M}$ 所表示的变换中经过原点的不变直线的方程。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

矩阵作用于列向量时，右边矩阵先作用。完整描述错切的固定轴与点的像，以及旋转的角度、方向和中心。

The transformations are applied in this order:

1. A shear with the $x$-axis fixed, mapping $(0,1)$ to $(2,1)$, represented by $\begin{pmatrix}1&2\\0&1\end{pmatrix}$.
2. An anticlockwise rotation through $\pi/4$ about the origin, represented by $\begin{pmatrix}\cos(\pi/4)&-\sin(\pi/4)\\\sin(\pi/4)&\cos(\pi/4)\end{pmatrix}$.

不变直线要求像点仍在同一直线上，不要求每点固定。设y=mx，比较像点斜率，另检查竖直线。

Multiplying the matrices,

For a point on $y=mx$, its image $(X,Y)$ is given by

The line is invariant when $Y=mX$, so

The vertical line $x=0$ is not invariant, since $(0,1)$ maps to $(\sqrt2/2,3\sqrt2/2)$, whose first coordinate is non-zero. Hence the two invariant lines through the origin are