Linear Transformations | 线性变换

📚 Linear Transformations | 线性变换

Linear transformations are a fundamental concept in A-Level Mathematics, bridging algebra and geometry through matrices. In this revision guide, we explore their definitions, matrix representations, common geometric transformations, composition, determinants, and invariants — all tailored to the AQA specification.

线性变换是A-Level数学中的一个核心概念，它通过矩阵将代数与几何紧密联系起来。在本复习指南中，我们将深入探讨其定义、矩阵表示、常见几何变换、复合变换、行列式以及不变性质——所有内容均针对AQA考试大纲量身定制。

1. What is a Linear Transformation? | 什么是线性变换？

A transformation T from a vector space to itself is linear if it preserves vector addition and scalar multiplication. Formally, for vectors u and v and a scalar c, we require T(u + v) = T(u) + T(v) and T(cu) = cT(u).

一个变换 T 如果保持向量加法和标量乘法，则称为线性变换。形式上，对于向量 u 和 v 以及标量 c，必须满足 T(u + v) = T(u) + T(v) 以及 T(cu) = cT(u)。

In practice, every linear transformation from R² to R² can be represented by a 2×2 matrix. The image of a point (x, y) is obtained by multiplying the matrix by the column vector (x, y).

实际上，从R²到R²的每一个线性变换都可以用一个2×2矩阵来表示。点 (x, y) 的像通过将矩阵与列向量 (x, y) 相乘得到。

2. Matrix Representation | 矩阵表示

For a 2D linear transformation, the matrix A = [[a, b], [c, d]] acts on a point (x, y) as follows:

对于二维线性变换，矩阵 A = [[a, b], [c, d]] 对点 (x, y) 的作用如下：

[x’] [a b] [x]
[y’] = [c d] [y]

where (x’, y’) is the image of (x, y). The columns of the matrix are the images of the standard basis vectors (1, 0) and (0, 1). This is a key property: the first column is T(1, 0), the second is T(0, 1).

其中 (x’, y’) 是 (x, y) 的像。矩阵的列是标准基向量 (1, 0) 和 (0, 1) 的像。这是一个关键性质：第一列为 T(1, 0)，第二列为 T(0, 1)。

For example, the transformation that maps (1, 0) to (2, 3) and (0, 1) to (‑1, 4) has matrix [[2, ‑1], [3, 4]].

例如，将 (1, 0) 映射到 (2, 3)、将 (0, 1) 映射到 (‑1, 4) 的变换，其矩阵为 [[2, ‑1], [3, 4]]。

3. Rotation | 旋转变换

A rotation about the origin by an angle θ anticlockwise is represented by:

绕原点逆时针旋转角度 θ 的变换矩阵为：

[[cos θ, ‑sin θ], [sin θ, cos θ]]

The image of (x, y) is (x cos θ – y sin θ, x sin θ + y cos θ). Common angles appear frequently in exams: 90°, 180°, 270°, and 60°.

点 (x, y) 的像为 (x cos θ – y sin θ, x sin θ + y cos θ)。常用角度如90°、180°、270°和60°在考试中经常出现。

R(90°) = [[0, ‑1], [1, 0]]
R(180°) = [[‑1, 0], [0, ‑1]]
R(270°) = [[0, 1], [‑1, 0]]

Note that clockwise rotation is obtained by replacing θ with –θ, which is also the inverse of the anticlockwise rotation.

注意，顺时针旋转可通过将 θ 替换为 –θ 得到，它也是逆时针旋转的逆变换。

4. Reflection | 反射变换

Reflection in a line through the origin can be represented using the angle θ that the line makes with the positive x-axis. The matrix is:

关于过原点且与x轴正方向成角 θ 的直线的反射矩阵为：

[[cos 2θ, sin 2θ], [sin 2θ, ‑cos 2θ]]

Special cases you must memorise:

你必须牢记的特殊情形：

Reflection in x-axis: [[1, 0], [0, ‑1]]

关于x轴的反射：[[1, 0], [0, ‑1]]
Reflection in y-axis: [[‑1, 0], [0, 1]]

关于y轴的反射：[[‑1, 0], [0, 1]]
Reflection in y = x: [[0, 1], [1, 0]]

关于直线 y = x 的反射：[[0, 1], [1, 0]]
Reflection in y = –x: [[0, ‑1], [‑1, 0]]

关于直线 y = –x 的反射：[[0, ‑1], [‑1, 0]]

Reflection is an isometry (preserves distances) and has determinant –1.

反射是等距变换（保持距离），其行列式为 –1。

5. Enlargement and Scaling | 放大与缩放

An enlargement centred at the origin with scale factor k is represented by:

以原点为中心、比例因子为 k 的放大变换矩阵为：

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

This scales both coordinates equally. If k > 1, the shape grows; if 0 < k < 1, it shrinks. A negative k also rotates the shape by 180°.

该矩阵对两个坐标进行等比例缩放。若 k > 1，图形变大；若 0 < k < 1，图形缩小；负的 k 还会使图形旋转180°。

A more general scaling uses different factors along the x- and y-axes:

更一般的缩放可以沿x轴和y轴使用不同的因子：

[[a, 0], [0, d]]

The determinant of this matrix is ad, which gives the area scale factor of the transformation.

该矩阵的行列式为 ad，它给出了变换的面积比例因子。

6. Shear | 剪切变换

A shear parallel to the x-axis leaves the y-coordinate unchanged and shifts x by a factor k times y. Its matrix is:

平行于x轴的剪切保持y坐标不变，并将x坐标按因子 k 乘以y进行平移。其矩阵为：

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

Similarly, a shear parallel to the y-axis has matrix:

类似地，平行于y轴的剪切矩阵为：

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

Shears have determinant 1, meaning they preserve area, even though they distort shapes. The invariant line of a shear is the axis to which it is parallel.

剪切变换的行列式为1，意味着它保持面积不变，尽管会扭曲形状。剪切的不变直线是其平行的轴。

7. Composition of Transformations | 复合变换

Applying one transformation followed by another corresponds to multiplying their matrices. If transformation A is applied first, then transformation B, the combined matrix is BA (not AB). This order is crucial.

先施加一个变换，再施加另一个变换，对应于将它们对应的矩阵相乘。如果先施加变换 A，再施加变换 B，则复合矩阵为 BA（而不是 AB）。这个顺序至关重要。

For example, a rotation by 90° followed by a reflection in the y-axis gives:

例如，先绕原点旋转90°，再关于y轴反射，得到：

[[‑1, 0], [0, 1]] × [[0, ‑1], [1, 0]] = [[0, 1], [1, 0]]

The result is a reflection in the line y = x. In exams, always write the matrices in the correct order and multiply carefully.

结果是一个关于直线 y = x 的反射。在考试中，务必按正确顺序写出矩阵并仔细相乘。

When combining more than two transformations, extend the principle: if transformations T₁, T₂, T₃ are applied in that order, the single equivalent matrix is T₃T₂T₁.

当组合两个以上变换时，扩展该原则：若按顺序 T₁、T₂、T₃ 施加，则等效的单一矩阵为 T₃T₂T₁。

8. Inverse Transformations | 逆变换

The inverse of a matrix A = [[a, b], [c, d]] is given by:

矩阵 A = [[a, b], [c, d]] 的逆矩阵为：

A⁻¹ = (1 / (ad – bc)) [[d, ‑b], [‑c, a]]

provided that the determinant Δ = ad – bc ≠ 0. If Δ = 0, the transformation maps all points onto a line or a point, and it has no inverse; such a transformation is called singular.

前提是行列式 Δ = ad – bc ≠ 0。若 Δ = 0，则变换将所有点映射到一条直线或一个点，且没有逆变换；这种变换称为奇异变换。

The inverse transformation undoes the original. For example, if A is a rotation by 90° anticlockwise, its inverse is a rotation by 90° clockwise, which is the same as the transpose of the rotation matrix (for rotations).

逆变换可以还原原始变换。例如，如果 A 是逆时针旋转90°，其逆就是顺时针旋转90°，对于旋转矩阵而言，这也等于其转置。

To verify that matrices are inverses, multiply them in both orders to get the identity matrix [[1, 0], [0, 1]].

要验证两个矩阵互为逆矩阵，可将其按两种顺序相乘，结果均为单位矩阵 [[1, 0], [0, 1]]。

9. Determinant and Area Scale Factor | 行列式与面积比例因子

For a 2×2 matrix A = [[a, b], [c, d]], the determinant is Δ = ad – bc. The absolute value of the determinant gives the area scale factor of the transformation.

对于2×2矩阵 A = [[a, b], [c, d]]，行列式为 Δ = ad – bc。行列式的绝对值给出了变换的面积比例因子。

If a shape has area S, then after transformation its area is |Δ| × S. If Δ is negative, the orientation of the shape is reversed (e.g., a reflection flips orientation).

若一个图形面积为 S，则变换后面积为 |Δ| × S。若 Δ 为负，则图形的方向发生反转（例如反射会翻转方向）。

For transformations in 3D, the determinant of the 3×3 matrix gives the volume scale factor. The same idea extends naturally.

对于三维变换，3×3矩阵的行列式给出体积比例因子。这一思想可以自然推广。

Area after = |det(A)| × Area before

10. Invariant Points and Lines | 不变点与不变直线

An invariant point is a point that is mapped to itself by the transformation. For a matrix A, we solve Ax = x, i.e., (A – I)x = 0.

不变点是变换后映射到自身的点。对于矩阵 A，我们求解 Ax = x，即 (A – I)x = 0。

For example, every reflection has a line of invariant points — the mirror line. The origin is always invariant for any linear transformation because A(0, 0) = (0, 0).

例如，每个反射都有一条不变点的直线——即镜面线。原点对于任何线性变换总是不变点，因为 A(0, 0) = (0, 0)。

An invariant line is a line that maps to itself as a whole (not necessarily pointwise). For a line through the origin, an eigenvector corresponding to eigenvalue 1 spans an invariant line. In shear transformations, the axis is a line of invariant points.

不变直线是一条映射到自身的直线（不一定逐点不变）。对于过原点的直线，特征值为1对应的特征向量张成一条不变直线。在剪切变换中，轴是一条不变点的直线。

To find invariant lines of the form y = mx + c, substitute into the transformation equations and solve for m and c, remembering that for linear transformations only lines through the origin can be invariant unless the transformation has a translation part (not covered here).

要求形如 y = mx + c 的不变直线，可将其代入变换方程并求解 m 和 c。注意，对于线性变换，只有过原点的直线才可能是不变直线（除非变换还包含平移，此处不讨论）。

11. Transformations in 3D (A-Level Further) | 三维变换（进阶数学）

For 3D linear transformations, we use 3×3 matrices. The columns represent the images of the standard basis vectors (1,0,0), (0,1,0), and (0,0,1).

对于三维线性变换，我们使用3×3矩阵。各列代表标准基向量 (1,0,0)、(0,1,0) 和 (0,0,1) 的像。

Common 3D transformations include rotations about the x-, y-, and z-axes. For example, rotation about the z-axis by angle θ uses:

常见三维变换包括绕x轴、y轴和z轴的旋转。例如，绕z轴旋转角度 θ 的矩阵为：

[[cos θ, ‑sin θ, 0], [sin θ, cos θ, 0], [0, 0, 1]]

Rotations about the x and y axes follow analogous patterns, with the identity row and column placed accordingly.

绕x轴和y轴的旋转遵循类似模式，只需将单位行和列放置在相应位置。

Determinants in 3D give the volume scale factor. A reflection in 3D has determinant –1, while a rotation has determinant +1.

三维行列式给出体积比例因子。三维反射的行列式为 –1，而旋转的行列式为 +1。

12. Summary and Exam Tips | 总结与考试技巧

To succeed with linear transformations in the AQA exam, follow these key steps:

要在AQA考试中成功应对线性变换，请遵循以下关键步骤：

Always identify the images of the standard basis vectors when constructing a matrix.

构造矩阵时，始终找出标准基向量的像。
Memorise the matrices for rotation, reflection, enlargement, and shear.

牢记旋转、反射、放大和剪切的矩阵。
Remember the order of composition: later transformations multiply on the left.

记住复合变换的顺序：后施加的变换左乘。
Use the determinant to find area/volume scale factors, and note sign changes for orientation.

利用行列式求面积/体积比例因子，并注意符号变化表示方向反转。
When finding invariant lines or points, set up equations and solve systematically.

求不变直线或不变点时，系统性地建立方程并求解。
Check your inverse matrices by multiplying them to get the identity.

通过相乘得到单位矩阵来验证逆矩阵。

Linear transformations are a high-yield topic. With practice, you can confidently handle both calculation and interpretation questions.

线性变换是高分考点。通过练习，你将能够自信地处理计算题和解释题。

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Linear Transformations | 线性变换