📚 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)。
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]]
ne 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 – s 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 – . 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）。
erminant 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。
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）
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。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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