📚 Matrix Transformations in 2D | 二维矩阵变换
ers; it represents a linear transformation of the plane. In AQA A-Level Mathematics, you need to recognise, apply and combine the standard matrix transformations in two dimensions.
2×2 矩阵不仅是数字表格；它表示平面的一种线性变换。在 AQA 数学 A-Level 中，你需要识别、应用并组合二维标准矩阵变换。
1. Linear Transformations and Matrices | 线性变换与矩阵
A linear transformation in 2D maps every point (x, y) to a new point (x', y') using two linear equations:
二维线性变换把每个点 (x, y) 映射到新点 (x', y')，使用两个线性方程：
x' = ax + by, y' = cx + dy
These equations can be written compactly in matrix form as:
这些方程可以紧凑地写成矩阵形式：
( x' ) = ( a b )( x )
( y' ) ( c d )( y )
The matrix M below is said to represent the transformation. The origin is always mapped to the origin.
下面的矩阵 M 表示该变换。原点总是映射到原点。
| a | b |
| c | d |
ines a linear transformation of the plane. This is why matrices are so useful in geometry.
每个 2×2 矩阵都定义了平面的一种线性变换，这正是矩阵在几何中非常实用的原因。
2. Matrix Multiplication and Column Vectors | 矩阵乘法与列向量
To apply a transformation to a point, write the coordinates as a column vector and multiply on the left by the transformation matrix.
要应用变换，先把坐标写成列向量，然后用变换矩阵左乘。
For M = (a b; c d) and point (x, y), the image is:
对于 M = (a b; c d) 和点 (x, y)，像点为：
( x' ) = ( a b )( x ) = ( ax + by )
( y' ) ( c d )( y ) ( cx + dy )
The first row times the column gives the new x-coordinate, and the second row times the column gives the new y-coordinate.
第一行乘列向量得到新的 x 坐标，第二行乘列向量得到新的 y 坐标。
This rule is the same for every point, so a single matrix represents the entire transformation.
对每个点都使用同样的规则，因此一个矩阵就可以表示整个变换。
3. The Identity Matrix | 单位矩阵
The identity matrix I = (1 0; 0 1) leaves every point unchanged. It plays the same role as the number 1 in multiplication.
单位矩阵 I = (1 0; 0 1) 使每个点保持不变。它的作用与乘法中的数字 1 相同。
I(x, y) = (x, y)
For any transformation matrix M, we have MI = IM = M.
对任意变换矩阵 M，都有 MI = IM = M。
Recognising the identity matrix helps you check whether a matrix has no geometric effect.
识别单位矩阵有助于判断一个矩阵是否没有几何效果。
4. Rotations About the Origin | 绕原点旋转
A rotation through angle θ anticlockwise about the origin is represented by:
绕原点逆时针旋转 θ 角的变换矩阵为：
R(θ) = ( cos θ -sin θ )
( sin θ cos θ )
This comes from sending the point (1, 0) to (cos θ, sin θ) and the point (0, 1) to (−sin θ, cos θ).
这是因为点 (1, 0) 被映射到 (cos θ, sin θ)，点 (0, 1) 被映射到 (−sin θ, cos θ)。
Important special cases are 90°, 180° and 270° anticlockwise rotations.
重要的特殊情况包括 90°、180° 和 270° 逆时针旋转。
- 90° anticlockwise: (0 −1; 1 0) | 90° 逆时针：(0 −1; 1 0)
- 180°: (−1 0; 0 −1) | 180°：(−1 0; 0 −1)
- 270° anticlockwise: (0 1; −1 0) | 270° 逆时针：(0 1; −1 0)
A rotation preserves both distances and orientation, and its determinant is always 1.
旋转保持距离和方向不变，其行列式始终为 1。
5. Reflections in Lines Through the Origin | 关于过原点直线的反射
Reflections in the coordinate axes and the lines y = x and y = −x are standard matrices.
关于坐标轴以及直线 y = x 和 y = −x 的反射是标准矩阵。
| Reflection line | Matrix | 反射线 |
|---|---|---|
| x-axis | (1 0; 0 −1) | x 轴 |
| y-axis | (−1 0; 0 1) | y 轴 |
| y = x | (0 1; 1 0) | y = x |
| y = −x | (0 −1; −1 0) | y = −x |
A reflection reverses orientation, so the determinant of any reflection matrix is −1.
反射会反转方向，因此任何反射矩阵的行列式都为 −1。
Points on the mirror line are mapped to themselves, so that line is a line of fixed points.
镜像线上的点映射到自身，因此该线是不动点直线。
6. Stretches and Enlargements | 拉伸与放大
A stretch parallel to the x-axis with scale factor k keeps y fixed and multiplies x by k. Its matrix is:
平行于 x 轴、比例因子为 k 的拉伸保持 y 不变，并将 x 乘以 k。其矩阵为：
