题目

Problem

Let $\mathbf A=\begin{pmatrix}1&a\\0&1\end{pmatrix}$ , where $a$ is a positive constant.

(a) State the type of the geometrical transformation in the $x$ – $y$ plane represented by $\mathbf A$ .

(1)

(b) Prove by mathematical induction that, for all positive integers $n$ ,

\mathbf A^n=\begin{pmatrix}1&na\\0&1\end{pmatrix}.

(5)

Let $\mathbf B=\begin{pmatrix}b&b\\a^{-1}&a^{-1}\end{pmatrix}$ , where $b$ is a positive constant.

(c) Find the equations of the invariant lines, through the origin, of the transformation in the $x$ – $y$ plane represented by $\mathbf A^n\mathbf B$ .

(6)

题目中文翻译

设 $\mathbf A=\begin{pmatrix}1&a\\0&1\end{pmatrix}$ ，其中 $a$ 是正常数。

(a) 写出 $\mathbf A$ 所表示的 $x$ – $y$ 平面内的几何变换的类型。

(1)

(b) 用数学归纳法证明，对于所有正整数 $n$ ，

\mathbf A^n=\begin{pmatrix}1&na\\0&1\end{pmatrix}.

(5)

设 $\mathbf B=\begin{pmatrix}b&b\\a^{-1}&a^{-1}\end{pmatrix}$ ，其中 $b$ 是正常数。

(6)

原卷通用作答要求

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。

解答

(a)

解法一

思路

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观察坐标变换：纵坐标不变，横坐标增加与纵坐标成比例的量。

答题过程

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The transformation is $(x,y)\mapsto(x+ay,y)$ . It is a $\boxed{\text{shear in the }x\text{-direction}}$ , with the $x$ -axis fixed and shear factor $a$ .

(b)

解法一

思路

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归纳法必须写出初始情形、明确的归纳假设、矩阵乘法及最终结论。

答题过程

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For $n=1$ ,

\mathbf A^1=\begin{pmatrix}1&a\\0&1\end{pmatrix},

so the formula holds.

Assume that for some positive integer $k$ ,

\mathbf A^k=\begin{pmatrix}1&ka\\0&1\end{pmatrix}.

Then

\begin{align*} \mathbf A^{k+1} =&\,\begin{pmatrix}1&ka\\0&1\end{pmatrix}\begin{pmatrix}1&a\\0&1\end{pmatrix}\\[4mm] =&\,\begin{pmatrix}1&a+ka\\0&1\end{pmatrix}\\[4mm] =&\,\begin{pmatrix}1&(k+1)a\\0&1\end{pmatrix}. \end{align*}

Thus the formula holds for $k+1$ whenever it holds for $k$ . By mathematical induction,

\boxed{\mathbf A^n=\begin{pmatrix}1&na\\0&1\end{pmatrix}}

for every positive integer $n$ .

CIE 9231 2022 June Paper 13 Q5

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程