📚 Matrix Transformations (2-Dimensional) | 二维矩阵变换
rix represents a linear transformation of the Cartesian plane. This topic requires you to interpret the geometric effect of a matrix, construct matrices for standard transformations, combine transformations in the correct order, and use determinants to describe area changes.
在 AQA A-Level 数学中，2×2 矩阵表示笛卡尔平面上的一个线性变换。本主题要求你解释矩阵的几何效果、构造标准变换矩阵、按正确顺序复合并用行列式描述面积变化。
1. Linear Transformations and Position Vectors | 线性变换与位置向量
j = (0, 1). If M = [a b; c d] acts on (x, y), the image is given by matrix multiplication.
线性变换将原点映射到自身，将直线映射为直线，并保持平行性。任一位置向量 (x, y) 可写成 x i + y j，其中 i = (1, 0)，j = (0, 1)。若 M = [a b; c d] 作用于 (x, y)，其像由矩阵乘法给出。
(x′, y′) = (a x + b y, c x + d y)
The numbers a and c tell us where the x-basis vector i goes, while b and d tell us where the y-basis vector j goes. This is the key idea behind all 2-dimensional matrix transformations.
数字 a 和 c 告诉我们 x 基向量 i 的像，数字 b 和 d 告诉我们 y 基向量 j 的像。这是所有二维矩阵变换的核心思想。
2. The Unit Square and Basis Vectors | 单位正方形与基向量
e unit square with vertices (0, 0), (1, 0), (1, 1), (0, 1) maps to a parallelogram with vertices (0, 0), (a, c), (a + b, c + d), (b, d).
矩阵 M 的列就是标准基向量的像。第一列是 M(1, 0) = (a, c)，第二列是 M(0, 1) = (b, d)。因此，顶点为 (0, 0)、(1, 0)、(1, 1)、(0, 1) 的单位正方形映射为顶点为 (0, 0)、(a, c)、(a + b, c + d)、(b, d) 的平行四边形。
This parallelogram gives a visual way to understand what a matrix does. If you can see where the unit square goes, you can describe the transformation precisely.
这个平行四边形为理解矩阵的作用提供了直观方法。如果你能看出单位正方形变成了什么形状，就能精确描述该变换。
3. Reading a Transformation from Its Matrix | 从矩阵读出变换
Look at the columns of the matrix to identify the transformation. If the columns are perpendicular unit vectors, the matrix is a rotation or a reflection. If one basis vector is unchanged and the other is shifted, it is a shear. If the basis vectors are scaled independently, it is a stretch.
观察矩阵的列可以识别变换类型。如果两列是互相垂直的单位向量，该矩阵是旋转或反射。如果一个基向量不变而另一个被平移，则是剪切。如果基向量被独立缩放，则是拉伸。
s a two-way stretch, not an enlargement, because the scale factors in the x- and y-directions are different.
例如，M = [2 0; 0 3] 将 i = (1, 0) 映射到 (2, 0)，将 j = (0, 1) 映射到 (0, 3)。这是一个双向拉伸，而不是放大，因为 x 方向和 y 方向的比例因子不同。
4. Common Transformation Matrices | 常见变换矩阵
matrices for AQA exams. The table below summarises the most common cases; k denotes a non-zero scale factor.
你应该记住 AQA 考试中常见的标准 2×2 矩阵。下表总结了最常见的情况；k 表示非零比例因子。
| Transformation | Matrix | Effect |
| Identity | [1 0; 0 1] | No change |
| Reflection in x-axis | [1 0; 0 -1] | i unchanged, j reflected |
| Reflection in y-axis | [-1 0; 0 1] | i reflected, j unchanged |
| Reflection in y = x | [0 1; 1 0] | Coordinates swapped |
| Reflection in y = -x | [0 -1; -1 0] | Coordinates swapped and negated |
| Rotation 90° anticlockwise | [0 -1; 1 0] | Quarter turn about origin |
| Rotation 180° | [-1 0; 0 -1] | Half turn about origin |
| Rotation 270° anticlockwise | [0 1; -1 0] | Three-quarter turn, or 90° clockwise |
| Stretch parallel to x-axis, factor k | [k 0; 0 1] | x-coordinates multiplied by k |
| Stretch parallel to y-axis, factor k | [1 0; 0 k] | y-coordinates multiplied by k |
| Shear parallel to x-axis, factor k | [1 k; 0 1] | x-coordinate becomes x + ky |
| Shear parallel to y-axis, factor k | [1 0; k 1] | y-coordinate becomes y + kx |
| Enlargement, centre origin, factor k | [k 0; 0 k] | Uniform scaling by k |
Learning these matrices by heart will save time in the exam, but you should also understand how each one is built from the images of i and j.
熟记这些矩阵可以在考试中节省时间，但你也应理解每个矩阵是如何从 i 和 j 的像构造出来的。
5. Rotations About the Origin | 绕原点的旋转
first column, and j = (0, 1) rotates to (-sin θ, cos θ), which becomes the second column.
绕原点逆时针旋转角度 θ 的一般矩阵为 [cos θ -sin θ; sin θ cos θ]。其原理是 i = (1, 0) 旋转到 (cos θ, sin θ)，成为第一列；j = (0, 1) 旋转到 (-sin θ, cos θ)，成为第二列。
0 1; -1 0]. A clockwise rotation is simply a negative angle, so 90° clockwise is θ = -90°.
对于特殊角，此公式给出标准矩阵：θ = 90° 得到 [0 -1; 1 0]，θ = 180° 得到 [-1 0; 0 -1]，θ = 270° 得到 [0 1; -1 0]。顺时针旋转只是负角度，因此顺时针 90° 就是 θ = -90°。
6. Reflections in Lines Through the Origin | 关于过原点直线的反射
For a line through the origin that makes an angle θ with the positive x-axis, the reflection matrix is [cos 2θ sin 2θ; sin 2θ -cos 2θ]. You can check this against the special cases in the standard table.
对于过原点且与正 x 轴夹角为 θ 的直线，反射矩阵为 [cos 2θ sin 2θ; sin 2θ -cos 2θ]。你可以将此公式与标准表中的特殊情况核对。
y = x, giving [0 1; 1 0]. When θ = 90°, the mirror line is the y-axis, giving [-1 0; 0 1].
当 θ = 0 时，镜线是 x 轴，得到 [1 0; 0 -1]。当 θ = 45° 时，镜线是 y = x，得到 [0 1; 1 0]。当 θ = 90° 时，镜线是 y 轴，得到 [-1 0; 0 1]。
7. Stretches and Shears | 拉伸与剪切
A stretch with scale factor k parallel to the x-axis acts as (x, y) ↦ (k x, y). Its matrix is [k 0; 0 1]. A stretch parallel to the y-axis acts as (x, y) ↦ (x, k y), with matrix [1 0; 0 k].
平行于 x 轴、比例因子为 k 的拉伸将 (x, y) 映射为 (k x, y)，其矩阵为 [k 0; 0 1]。平行于 y 轴、比例因子为 k 的拉伸将 (x, y) 映射为 (x, k y)，其矩阵为 [1 0; 0 k]。
A shear parallel to the x-axis with factor k maps (x, y) ↦ (x
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