📚 Mastering Linear Transformations in A-Level Maths | 精通 A-Level 数学中的线性变换
ransformation of the plane is linear when it fixes the origin and preserves the vector operations of addition and scalar multiplication, and every such transformation can be written in the compact form T(x) = Mx with a 2 × 2 matrix M. This article builds the complete exam toolkit: representing rotations, reflections, stretches and shears by matrices, composing transformations in the correct order, finding inverse transformations, interpreting the determinant as an area scale factor, and locating invariant points and invariant lines. Throughout, matrices are written as [a b; c d], where the semicolon separates the first row from the second row.
线性变换是 A-Level 进阶数学中最有价值、也最容易出高分题的主题之一。当一个平面变换把原点固定不动，并且保持向量的加法与数乘运算时，它就是线性变换；而每一个这样的变换都可以写成紧凑形式 T(x) = Mx，其中 M 是一个 2 × 2 矩阵。本文将系统构建完整的应试工具箱：用矩阵表示旋转、反射、拉伸与剪切；按正确顺序复合变换；求逆变换；把行列式解释为面积比例因子；以及寻找不变点与不变直线。全文中的矩阵均写作 [a b; c d]，分号用来分隔第一行与第二行。
1. What is a Linear Transformation? | 什么是线性变换？
and T(kx) = kT(x). Together these properties mean that T is completely determined by what it does to the two basis vectors, and that the origin is always fixed, since T(0) = T(0 · x) = 0 · T(x) = 0.
如果平面上的变换 T 对任意位置向量 x、y 以及任意实数 k 都满足两个条件，就称为线性变换：T(x + y) = T(x) + T(y) 以及 T(kx) = kT(x)。这两条性质合起来说明：T 完全由它对两个基向量的作用所决定；而且原点永远不动，因为 T(0) = T(0 · x) = 0 · T(x) = 0。
ple: a translation sends the origin to a different point, so it fails the test immediately. Any transformation that moves the origin, such as a glide or a general translation, is therefore not linear and cannot be represented by a 2 × 2 matrix in this way.
旋转、反射、以原点为中心的放大、单向拉伸与剪切都是线性变换。平移是经典的例外：平移把原点送到另一个点，因此立刻不满足条件。任何移动原点的变换（例如滑移或一般平移）都不是线性变换，也就无法用这种 2 × 2 矩阵表示。
T(x) = Mx and T(0) = 0
2. Representing Transformations with 2 × 2 Matrices | 用 2 × 2 矩阵表示变换
vector. The key insight is that the columns of M are simply the images of the two basis vectors: the first column is the image of (1, 0) and the second column is the image of (0, 1). If you know where the unit square goes, you know the matrix.
平面上每一个线性变换都可以写成 T(x) = Mx，其中 M 是固定的 2 × 2 矩阵，x 是列向量。关键洞察在于：M 的两列就是两个基向量的像——第一列是 (1, 0) 的像，第二列是 (0, 1) 的像。只要知道单位正方形去了哪里，就知道了矩阵。
1 + 3 × 2) = (4, 6). Working backwards, if a question tells you that (1, 0) maps to (0, 1) and (0, 1) maps to (−1, 0), you can write the matrix straight down as [0 −1; 1 0].
要求一般点的像，只需用矩阵乘以它的位置向量。例如 M = [2 1; 0 3] 时，点 (1, 2) 映射到 (2 × 1 + 1 × 2, 0 × 1 + 3 × 2) = (4, 6)。反过来，如果题目告诉你 (1, 0) 映射到 (0, 1)，而 (0, 1) 映射到 (−1, 0)，你可以直接把矩阵写出来：[0 −1; 1 0]。
3. The Standard Transformation Matrices You Must Memorise | 必须背下来的标准变换矩阵
Exam boards expect instant recall of the standard matrices. The table below lists the ones that appear most often, together with their determinants, which you should also be able to state without calculation.
考试局要求你能瞬间回忆出这些标准矩阵。下表列出了最常出现的几个，并附上它们的行列式——这些行列式你也应当能够脱口而出，而无需计算。
| Transformation 变换 | Matrix 矩阵 | det M |
| Anticlockwise rotation by θ 逆时针旋转 θ | [cos θ −sin θ; sin θ cos θ] | 1 |
| Reflection in the x-axis 关于 x 轴反射 | [1 0; 0 −1] | −1 |
| Reflection in the y-axis 关于 y 轴反射 | [−1 0; 0 1] | −1 |
| Reflection in y = x 关于 y = x 反射 | [0 1; 1 0] | −1 |
| Reflection in y = −x 关于 y = −x 反射 | [0 −1; −1 0] | −1 |
| Stretch by factor k parallel to x 沿 x 方向以 k 拉伸 | [k 0; 0 1] | k |
| Stretch by factor k parallel to y 沿 y 方向以 k 拉伸 | [1 0; 0 k] | k |
| Enlargement by factor k 以 k 放大 | [k 0; 0 k] | k² |
| Shear parallel to x 沿 x 方向剪切 | [1 k; 0 1] | 1 |
nd stretches and enlargements change area by the stated factor. A determinant of zero would mean the transformation collapses the plane onto a line, which is why it can never be invertible.
请注意行列式的规律：旋转完全保持面积（det = 1）；反射保持面积大小但翻转取向（det = −1）；拉伸与放大则按相应因子改变面积。行列式为零意味着变换把整个平面压扁成一条直线，这也是它绝不可逆的原因。
4. Rotation Matrices and Angle Conventions | 旋转矩阵与角度约定
: positive angles are anticlockwise. A clockwise rotation through θ is the same as an anticlockwise rotation through −θ, so you simply replace θ by −θ and use the identities cos(−θ) = cos θ and sin(−θ) = −sin θ.
旋转矩阵 R(θ) = [cos θ −sin θ; sin θ cos θ] 把每个点绕原点逆时针旋转角度 θ。这个约定至关重要：正角代表逆时针。顺时针旋转 θ 等价于逆时针旋转 −θ，因此只需把 θ 换成 −θ，并利用恒等式 cos(−θ) = cos θ 与 sin(−θ) = −sin θ。
A rotation through 90° therefore gives [0 −1; 1 0], and a rotation through 180° gives [−1 0; 0 −1], which is the same as an enlargement by factor −1. Rotations form a family in which the order of composition never matters: rotating by α and then by β always equals rotating by α + β, which is a useful check on your arithmetic in multi-step problems.
因此旋转 90° 得到 [0 −1; 1 0]，旋转 180° 得到 [−1 0; 0 −1]，这与以因子 −1 放大完全相同。旋转这一族变换的复合顺序永远无所谓：先旋转 α 再旋转 β 总是等于旋转 α + β——这在多步问题中是检验计算的好办法。
R(α)R(β) = R(β)R(α) = R(α + β)
5. Reflections, Stretches and Shears in Detail | 反射、拉伸与剪切的细节
on matrix. You can verify this quickly as an exam check. Stretches act in one direction only, leaving the perpendicular direction untouched, so a stretch parallel to the x-axis fixes every point on the x-axis while moving every other point horizontally.
反射的逆变换就是它自己：对同一条直线反射两次，每个点都会回到原处，因此每个反射矩阵都满足 M² = I。考试中可以用这一点快速验算。拉伸只作用在一个方向上，垂直方向不受影响，所以沿 x 方向的拉伸会固定 x 轴上的每一个点，而让其他点水平移动。
A shear parallel to the x-axis, given by [1 k; 0 1], slides each point horizontally by an amount proportional to its height. The shear is the only standard transformation that changes the shape of the unit square into a parallelogram without altering its area, which is exactly why its determinant equals 1. Recognising a shear from a diagram, where vertical lines stay vertical but slant into parallelograms, saves valuable time in the exam.
沿 x 方向的剪切由 [1 k; 0 1] 给出，它让每个点水平滑动，滑动量与它的高度成正比。剪切是唯一一种把单位正方形变成平行四边形、却不改变其面积的标准变换——这正是它的行列式等于 1 的原因。从图形中识别剪切（竖线仍然保持竖直，但整体斜成平行四边形）能在考试中节省宝贵时间。
6. Composing Transformations in the Correct Order | 按正确顺序复合变换
The single most common error in this topic is composing in the wrong order. If transformation T₁ is applied first and T₂ second, the combined transformation is T₂T₁, represented by the matrix product M₂M₁. The matrix nearest the vector acts first, so the order is right-to-left when you read a product.
本主题中最常见的错误就是复合顺序颠倒。如果先做变换 T₁、再做 T₂，那么合成变换是 T₂T₁，由矩阵乘积 M₂M₁ 表示。最靠近向量的矩阵最先作用，所以读乘积时要按从右到左的顺序。
he x-axis and then rotating 90° anticlockwise gives R(90°)M_reflection, whereas rotating first and then reflecting gives M_reflection·R(90°), and these two results are certainly different matrices. Because the associative law does hold, you may multiply three or more matrices in any bracketing you like: (M₃M₂)M₁ = M₃(M₂M₁).
矩阵乘法一般不可交换，所以 M₂M₁ 与 M₁M₂ 通常描述不同的变换。例如，先关于 x 轴反射再逆时针旋转 90° 得到 R(90°)M_reflection，而先旋转再反射得到 M_reflection·R(90°)，这两个结果显然是不同的矩阵。由于结合律成立，三个或更多矩阵相乘时你可以任意加括号：(M₃M₂)M₁ = M₃(M₂M₁)。
When a question gives you transformations in words, translate each one into its matrix first, then decide the order carefully from the wording. Phrases such as 'followed by', 'and then' and 'after which' indicate that the transformation just mentioned acts second, and therefore sits on the left of the product.
当题目用文字给出变换时，先把每一个转换成矩阵，再根据措辞仔细判断顺序。诸如 "followed by"、"and then"、"after which" 这类说法都表示刚刚提到的变换是第二步作用，因此它要放在乘积的左边。
7. Inverse Transformations | 逆变换
ies, negating the other two, and dividing by the determinant. Writing M = [a b; c d], we obtain M⁻¹ = (1/(ad − bc))[d −b; −c a].
一个线性变换可逆的充要条件就是它的行列式不为零。求逆矩阵的方法是：交换主对角线上的元素，另两个元素变号，再除以行列式。记 M = [a b; c d]，则 M⁻¹ = (1/(ad − bc))[d −b; −c a]。
negative angle; for a reflection the inverse is the same reflection; for a shear the inverse is the shear with k replaced by −k. If det M = 0 the transformation is singular, the unit square collapses onto a line segment, and no inverse exists because information has been lost.
逆变换在几何上就是"撤销"该变换：如果 T 把 A 映射到 B，那么 T⁻¹ 把 B 映射回 A。旋转的逆是旋转负角；反射的逆是同一条反射；剪切的逆是把 k 换成 −k 的剪切。如果 det M = 0，变换是奇异的，单位正方形被压成一条线段，信息已经丢失，因此不存在逆变换。
MM⁻¹ = M⁻¹M = I, det M ≠ 0
8. The Determinant as an Area Scale Factor | 行列式作为面积比例因子
, the absolute value of the determinant is the factor by which areas are multiplied. A unit square of area 1 becomes a parallelogram of area |det M|, so a triangle of area 6 becomes a triangle of area 6|det M|, and the same applies to any region.
对于矩阵 M = [a b; c d]，其行列式是 det M = ad − bc。几何上，行列式的绝对值就是面积被乘上的倍数。面积为 1 的单位正方形变成面积为 |det M| 的平行四边形；因此面积为 6 的三角形变成面积为 6|det M| 的三角形，对任何区域都成立。
The sign of the determinant carries its own meaning: a negative determinant indicates that the orientation of the plane has been reversed, turning a clockwise labelling of vertices anticlockwise. This is exactly what happens under a reflection. Questions frequently ask for the area of an image, or for the value of k that makes a transformation halve the area, and both are answered directly from |det M|.
行列式的符号本身也有含义：负的行列式说明平面的取向被翻转，原来顺时针排列的顶点变成逆时针。反射正是如此。考题经常问像的面积，或者问使某变换把面积减半的 k 值，这两类问题都可以直接用 |det M| 回答。
Area of image = |det M| × Area of object
9. Invariant Points | 不变点
variant point, and it is often the only one. To find all invariant points, solve the equation Mx = x, which rearranges to (M − I)x = 0, where I is the 2 × 2 identity matrix.
变换的不变点就是映射到自身的点。由于每个线性变换都固定原点，原点永远是不变点，而且常常是唯一的不变点。求所有不变点的方法，是解方程 Mx = x，整理后得到 (M − I)x = 0，其中 I 是 2 × 2 单位矩阵。
det(M − I) = 0, the two equations in the system are dependent and there is a whole line of invariant points passing through the origin. This second case arises whenever the transformation has an eigenvalue of 1, and recognising it immediately tells you that the answer is a line rather than a point.
解的个数取决于 M − I 的行列式。如果 det(M − I) ≠ 0，则唯一解是 x = 0，原点就是唯一的不变点。如果 det(M − I) = 0，方程组中两个方程线性相关，于是存在整条过原点的不变点直线。这第二种情形出现在变换具有特征值 1 的时候；一旦识别出来，你立刻就知道答案是直线而不是点。
2], whose determinant is 2. Since det(M − I) ≠ 0, the only invariant point is the origin.
示例：对 M = [2 1; 0 3]，构造 M − I = [1 1; 0 2]，其行列式为 2。因为 det(M − I) ≠ 0，唯一的不变点就是原点。
10. Invariant Lines | 不变直线
must pass through the origin, which is a handy first check: the equation will always have the form y = mx or x = 0.
不变直线是指作为整体集合映射到自身的直线，即使线上的个别点会移动。线性变换的每条不变直线都必须经过原点，这是个很好用的第一步检验：方程一定形如 y = mx 或 x = 0。
= m, which rearranges into the quadratic bm² + (a − d)m − c = 0. Solve this quadratic to find the gradients of all non-vertical invariant lines.
将一般点 (x, mx) 代入 M = [a b; c d] 得到像 (ax + bmx, cx + dmx)。像仍落在同一条直线上的充要条件是 (c + dm)/(a + bm) = m，整理后得到二次方程 bm² + (a − d)m − c = 0。解这个二次方程即可求出所有非竖直不变直线的斜率。
mage of (0, 1) is a multiple of (0, 1); if it is, the y-axis is also invariant. For a reflection in the line y = x, the quadratic becomes m² − 1 = 0, giving the
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