Matrix Transformations (2-Dimensional) | 矩阵变换（二维）

📚 Matrix Transformations (2-Dimensional) | 矩阵变换（二维）

Matrix transformations are a fundamental topic in A-Level Mathematics. A 2×2 matrix can be used to transform points in a two-dimensional plane, producing reflections, rotations, enlargements, and shears. This article provides a comprehensive guide to understanding and applying these transformations, with all results aligned with the AQA specification.

矩阵变换是A-Level数学中的一个基础专题。2×2矩阵可用于变换二维平面中的点，产生反射、旋转、放大和剪切等效果。本文将全面指导您理解和应用这些变换，所有结论均与AQA考纲保持一致。

1. Introduction to Matrix Transformations | 矩阵变换简介

A matrix transformation maps a point (x, y) to a new point (x’, y’) by multiplying a 2×2 matrix by a column vector. If M = [[a, b], [c, d]] and the point is represented as the column vector [x; y], then the image is given by the matrix product below.

矩阵变换通过将2×2矩阵与列向量相乘，将点(x, y)映射到新点(x’, y’)。若M = [[a, b], [c, d]]，点用列向量[x; y]表示，则像由下述矩阵乘积给出。

[x’; y’] = [[a, b], [c, d]] [x; y] = [ax + by; cx + dy]

Each column of the transformation matrix tells us where the basis vectors i = (1, 0) and j = (0, 1) are mapped. The first column (a, c) is the image of i, and the second column (b, d) is the image of j. This interpretation is the single most useful tool for both constructing and identifying transformation matrices.

变换矩阵的每一列告诉我们基向量i = (1, 0)和j = (0, 1)映射到哪里。第一列(a, c)是i的像，第二列(b, d)是j的像。这种按列解释的方法既是构造变换矩阵、也是识别变换矩阵时最有用的工具。

2. The General Form | 一般形式

For a general 2×2 matrix M = [[a, b], [c, d]], the transformation of a point (x, y) is expressed by two linear equations.

对于一般2×2矩阵M = [[a, b], [c, d]]，点(x, y)的变换由两个线性方程表示。

x’ = ax + by, y’ = cx + dy

Special cases include the zero matrix [[0, 0], [0, 0]], which maps every point to the origin, and the identity matrix I = [[1, 0], [0, 1]], which leaves every point unchanged. In general, the columns of the matrix are simply the images of the two coordinate axes’ unit vectors.

特殊情况包括零矩阵[[0, 0], [0, 0]]，它将所有点映射到原点；还有单位矩阵I = [[1, 0], [0, 1]]，它使每个点保持不变。一般而言，矩阵的列就是两个坐标轴单位向量的像。

Because the mapping is linear, straight lines are always mapped to straight lines, and the origin is always fixed. These properties distinguish matrix transformations from translations, which require an extended matrix form.

由于映射是线性的，直线总是映射为直线，原点始终保持不动。这些性质将矩阵变换与平移区分开来——平移需要扩展矩阵的形式。

3. Reflection Transformations | 反射变换

Reflections flip points across a straight line through the origin. The following standard reflection matrices must be memorised for the exam:

反射是使点关于过原点的直线翻转。以下标准反射矩阵是考试必须牢记的：

Reflection in the x-axis | 关于x轴的反射:

[[1, 0], [0, -1]]

Reflection in the y-axis | 关于y轴的反射:

[[-1, 0], [0, 1]]

Reflection in the line y = x | 关于直线y = x的反射:

[[0, 1], [1, 0]]

Reflection in the line y = -x | 关于直线y = -x的反射:

[[0, -1], [-1, 0]]

For a general reflection axis through the origin making an angle θ with the positive x-axis, the transformation matrix is given below. This formula can be used to derive any line-reflection matrix.

对于过原点且与x轴正方向成θ角的一般反射轴，其变换矩阵如下所示。该公式可用于推导任意直线反射的矩阵。

[[cos 2θ, sin 2θ], [sin 2θ, -cos 2θ]]

A reflection is its own inverse, since applying the same reflection twice returns every point to its original position. The determinant of any reflection matrix is -1, which reflects the fact that orientation is reversed.

反射是其自身的逆变换，因为连续应用两次相同的反射会使每个点回到原始位置。任何反射矩阵的行列式均为-1，这反映了方向被翻转的事实。

4. Rotation Transformations | 旋转变换

Rotations turn points about the origin through a given angle. For an anticlockwise rotation by angle θ, the transformation matrix is the following.

旋转是使点绕原点转过给定角度。对于逆时针旋转角度θ，变换矩阵如下。

[[cos θ, -sin θ], [sin θ, cos θ]]

The standard rotation matrices you must know for AQA are listed below.

AQA考试中必须掌握的标准旋转矩阵如下所列。

90° anticlockwise (θ = 90°): [[0, -1], [1, 0]] — 逆时针旋转90°
180° (θ = 180°): [[-1, 0], [0, -1]] — 旋转180°
270° anticlockwise (θ = 270°): [[0, 1], [-1, 0]] — 逆时针旋转270°

For a clockwise rotation, substitute a negative angle. For example, a clockwise rotation of 90° uses θ = -90°, producing the matrix [[0, 1], [-1, 0]]. The determinant of any rotation matrix is +1, confirming that rotations preserve both area and orientation.

对于顺时针旋转，代入负角度。例如，顺时针旋转90°使用θ = -90°，得到矩阵[[0, 1], [-1, 0]]。任何旋转矩阵的行列式均为+1，这确认了旋转保持面积和方向均不变。

5. Enlargement and Scaling | 放大与缩放

A scaling transformation multiplies the x- and y-coordinates by scale factors. The matrix for scaling by factor k in the x-direction and factor l in the y-direction is diagonal:

缩放变换将x坐标和y坐标分别乘以比例因子。在x方向缩放k倍、y方向缩放l倍的矩阵是对角矩阵：

[[k, 0], [0, l]]

If k = l = s, this is a uniform enlargement with scale factor s, which enlarges every length by the same factor and multiplies all areas by s². The matrix is [[s, 0], [0, s]].

若k = l = s，则为比例因子为s的均匀放大，所有长度按相同因子放大，所有面积乘以s²。其矩阵为[[s, 0], [0, s]]。

If k or l is negative, the enlargement includes a reflection in the corresponding axis. For example, [[-2, 0], [0, 3]] scales x by -2 (which reflects in the y-axis and enlarges by 2) and y by 3. A point (1, 1) is mapped to (-2, 3).

若k或l为负，则放大同时包含对应轴的反射。例如，[[-2, 0], [0, 3]]将x缩放-2倍（即关于y轴反射并放大2倍），将y缩放3倍。点(1, 1)被映射到(-2, 3)。

6. Shear Transformations | 剪切变换

A shear transformation shifts points parallel to one axis by an amount proportional to their coordinate on the other axis. A shear parallel to the x-axis with shear factor m has the following matrix.

剪切变换使点沿平行于某一轴的方向移动，移动量与它们在另一轴上的坐标成正比。沿x轴方向、剪切因子为m的剪切矩阵如下。

[[1, m], [0, 1]]

Similarly, a

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Matrix Transformations (2-Dimensional) | 矩阵变换（二维）