Matrix Transformations in 2D | 二维矩阵变换

📚 Matrix Transformations in 2D | 二维矩阵变换

A 2D matrix transformation uses a 2×2 matrix M to map a point (x, y) to a new point (x’, y’) by multiplying the position vector. This is a core topic in AQA A-Level Mathematics, where you need to identify, combine and invert geometric transformations represented by matrices.

二维矩阵变换使用一个2×2矩阵 M 将点 (x, y) 映射到新点 (x’, y’)，方法是乘以位置向量。这是 AQA A-Level 数学的核心主题，你需要识别、组合并求由矩阵表示的几何变换的逆变换。

1. What is a 2D Matrix Transformation? | 什么是二维矩阵变换？

A point (x, y) is written as a column vector [x; y]. A 2×2 matrix M = [a b; c d] acts on this vector to give the image vector [x’; y’] = M [x; y].

点 (x, y) 写作列向量 [x; y]。一个2×2矩阵 M = [a b; c d] 作用在该向量上，得到像向量 [x’; y’] = M [x; y]。

[ x’ ] [ a b ] [ x ]

[ y’ ] = [ c d ] [ y ]

This multiplication generates a linear transformation. Linear transformations map straight lines to straight lines, the origin to the origin, and preserve parallelism.

这种乘法产生一个线性变换。线性变换将直线映射为直线，将原点映射为原点，并保持平行性。

2. The Identity Matrix and Invariance | 单位矩阵与不变性

The identity matrix I = [1 0; 0 1] leaves every point unchanged. It is the matrix equivalent of a “do nothing” transformation.

单位矩阵 I = [1 0; 0 1] 使每个点保持不变。它相当于“什么都不做”的变换矩阵。

A point or line is called invariant if it is mapped to itself. For example, under reflection in the x-axis, every point on the x-axis is invariant.

如果点或线被映射到自身，则称其为不变的。例如，在关于 x 轴的反射下，x 轴上的每个点都是不变的。

To find if a line is invariant, apply the matrix to a general point on the line and check whether the image still satisfies the line equation.

要判断一条直线是否不变，可将矩阵作用在直线上的一个一般点，并检查像是否仍然满足直线方程。

3. Scaling (Enlargement) Transformations | 缩放变换

A uniform scaling by factor k is represented by the matrix [k 0; 0 k]. This enlarges or shrinks the shape by a factor k in all directions.

缩放因子为 k 的均匀缩放由矩阵 [k 0; 0 k] 表示。它使图形在所有方向上放大或缩小 k 倍。

A non-uniform scaling with factors p in the x-direction and q in the y-direction is given by [p 0; 0 q].

在 x 方向缩放 p 倍、y 方向缩放 q 倍的非均匀缩放由 [p 0; 0 q] 给出。

For example, applying [2 0; 0 3] to the point (1, 1) gives (2, 3).

例如，将 [2 0; 0 3] 作用于点 (1, 1) 得到 (2, 3)。

4. Reflection Transformations | 反射变换

Reflection in the x-axis: [1 0; 0 -1]. Reflection in the y-axis: [-1 0; 0 1].

关于 x 轴的反射：[1 0; 0 -1]。关于 y 轴的反射：[-1 0; 0 1]。

Reflection in the line y = x: [0 1; 1 0]. Reflection in the line y = -x: [0 -1; -1 0].

关于直线 y = x 的反射：[0 1; 1 0]。关于直线 y = -x 的反射：[0 -1; -1 0]。

These matrices satisfy M² = I, so applying the same reflection twice returns the original point.

这些矩阵满足 M² = I，所以连续应用两次相同的反射会回到原始点。

5. Rotation Transformations | 旋转变换

An anticlockwise rotation through angle θ about the origin is represented by the matrix [cosθ -sinθ; sinθ cosθ].

绕原点逆时针旋转角度 θ 由矩阵 [cosθ -sinθ; sinθ cosθ] 表示。

A clockwise rotation through θ is equivalent to an anticlockwise rotation through -θ, giving [cosθ sinθ; -sinθ cosθ].

顺时针旋转 θ 等价于逆时针旋转 -θ，得到 [cosθ sinθ; -sinθ cosθ]。

For example, a 90° anticlockwise rotation has matrix [0 -1; 1 0]. Under it, the point (1, 0) moves to (0, 1).

例如，90° 逆时针旋转的矩阵是 [0 -1; 1 0]。在该变换下，点 (1, 0) 移动到 (0, 1)。

6. Shear Transformations | 剪切变换

A shear parallel to the x-axis with shear factor k is given by [1 k; 0 1]. It moves each point horizontally by k times its y-coordinate.

平行于 x 轴、剪切因子为 k 的剪切由 [1 k; 0 1] 给出。它将每个点水平移动 k 乘以它的 y 坐标。

Similarly, a shear parallel to the y-axis is [1 0; k 1]. Points on the fixed line (axis of shear) remain unchanged.

类似地，平行于 y 轴的剪切是 [1 0; k 1]。固定线（剪切轴）上的点保持不变。

Shears have determinant 1, so they preserve areas.

剪切的行列式为 1，因此它们保持面积不变。

7. Composition of Transformations | 复合变换

If transformation A is applied first, followed by transformation B, the overall matrix is B A. Order matters: matrix multiplication is not commutative.

如果先应用变换 A，再应用变换 B，则总矩阵为 B A。顺序很重要：矩阵乘法不满足交换律。

Example: Reflect in the y-axis [ -1 0; 0 1 ], then rotate 90° anticlockwise [0 -1; 1 0]. The combined matrix is:

示例：先关于 y 轴反射 [ -1 0; 0 1 ]，再逆时针旋转 90° [0 -1; 1 0]。组合矩阵为：

[0 -1; 1 0] × [-1 0; 0 1] = [0 1; -1 0]

The same two transformations in the reverse order give a different result, showing the importance of order.

相同两个变换按相反顺序进行会得到不同结果，这说明了顺序的重要性。

8. Determinant and Area Scale Factor | 行列式与面积缩放因子

For M = [a b; c d], the determinant is det(M) = ad – bc. The absolute value |det(M)| is the area scale factor of the transformation.

对于 M = [a b; c d]，行列式为 det(M) = ad – bc。绝对值 |det(M)| 是变换的面积缩放因子。

If a shape has area A, its image has area A × |det(M)|. If det(M) = 0, the transformation collapses the plane onto a line, so the image area is 0.

如果图形面积为 A，则其像的面积为 A × |det(M)|。如果 det(M) = 0，则变换将平面压缩到一条直线上，因此像的面积为 0。

The sign of det(M) indicates orientation. A negative determinant means the transformation reverses orientation (e.g. a reflection).

det(M) 的符号表示方向。负的行列式意味着变换反转方向（例如反射）。

9. Inverse Transformations | 逆变换

For an invertible matrix M with det(M) ≠ 0, the inverse matrix is M⁻¹ = 1/(ad – bc) [d -b; -c a]. It undoes the effect of M.

对于 det(M) ≠ 0 的可逆矩阵 M，逆矩阵为 M⁻¹ = 1/(ad – bc) [d -b; -c a]。它撤销 M 的效果。

Applying M then M⁻¹ (or M⁻¹ then M) gives the identity: M⁻¹M = I.

先应用 M 再应用 M⁻¹（或先 M⁻¹ 再 M）得到单位阵：M⁻¹M = I。

Geometrically, the inverse matrix represents the reverse transformation. For example, the inverse of a 90° anticlockwise rotation is a 90° clockwise rotation.

从几何上讲，逆矩阵表示相反的变换。例如，90° 逆时针旋转的逆是 90° 顺时针旋转。

10. Using Matrices to Find Images of Points and Lines | 用矩阵求点和直线的像

To find the image of a point (x, y), simply multiply the matrix by the column vector [x; y].

要求点 (x, y) 的像，只需将矩阵乘以列向量 [x; y]。

To find the image of a line, write it in parametric form (x, y) = (x₀ + t a, y₀ + t b), apply the matrix to each position vector, then eliminate the parameter t.

要求直线的像，可将其写成参数形式 (x, y) = (x₀ + t a, y₀ + t b)，将矩阵作用于每个位置向量，然后消去参数 t。

Alternatively, find the images of two points on the line and determine the equation of the line through them.

另一种方法是求出直线上两个点的像，然后确定通过这两个像点的直线方程。

11. Exam-Style Worked Example | 考试风格例题

Let M = [2 -1; 3 4]. Find the image of the point (2, 3) under M, the area scale factor, and the inverse matrix.

设 M = [2 -1; 3 4]。求点 (2, 3) 在 M 下的像、面积缩放因子以及逆矩阵。

Solution: Image = M [2; 3] = [2×2 + (-1)×3; 3×2 + 4×3] = [1; 18]. So the image is (1, 18).

解：像 = M [2; 3] = [2×2 + (-1)×3; 3×2 + 4×3] = [1; 18]。所以像是 (1, 18)。

det(M) = 2×4 – (-1)×3 = 8 + 3 = 11, so the area scale factor is 11.

det(M) = 2×4 – (-1)×3 = 8 + 3 = 11，所以面积缩放因子为 11。

M⁻¹ = 1/11 [4 1; -3 2] = [4/11 1/11; -3/11 2/11].

M⁻¹ = 1/11 [4 1; -3 2] = [4/11 1/11; -3/11 2/11]。

12. Summary | 总结

2D matrix transformations are a powerful way to describe geometric operations. You must know the standard matrices for reflection, rotation, scaling and shear, understand composition order, use determinants for area scaling, and compute inverse matrices to reverse transformations.

二维矩阵变换是描述几何运算的强大工具。你必须掌握反射、旋转、缩放和剪切的标準矩阵，理解复合变换的顺序，使用行列式求面积缩放，并计算逆矩阵来反转变换。

Practice identifying transformations from given matrices and combining them becomes straightforward once you master the underlying linear algebra.

一旦你掌握了基础的线性代数，练习从给定矩阵识别变换以及组合变换就会变得简单直接。

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Matrix Transformations in 2D | 二维矩阵变换