Matrix Transformations in 2D | 二维矩阵变换

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

rices. When a column vector v = (x, y) is multiplied by a matrix M, the result Mv is the image of v under that transformation. This article covers the core AQA techniques for identifying, combining, inverting and interpreting 2 × 2 matrix transformations, including determinants, area scale factors, inverse matrices, invariant points and invariant lines.

在二维空间中，反射、旋转、放大和剪切等许多几何变换都可以用 2 × 2 矩阵高效表示。当一个列向量 v = (x, y) 乘以矩阵 M 时，得到的新列向量 Mv 就是原像在该变换下的像。本文涵盖 AQA 考试中关于识别、复合、求逆和解释二维矩阵变换的核心技巧，并系统讲解行列式、面积缩放因子、逆矩阵、不变点与不变直线。

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1. Linear Transformations and Matrices | 线性变换与矩阵

linear transformation T maps a general position vector (x, y) to a new position vector (x', y') where x' = ax + by and y' umn vector by multiplication, so the whole transformation is encoded in just four numbers.

线性变换 T 将一般位置向量 (x, y) 映射为新位置向量 (x', y')，其中 x' = ax + by，y' = cx + dy。2 × 2 矩阵 M = [a b; c d] 通过乘法作用于列向量，因此整个变换只由四个数就能完整编码。

Mv = [a b; c d] [x; y] = [ax + by; cx + dy]

line segment is still a line segment, and parallel lines remain parallel. The transformation is completely determined by what it does to the two unit vectors e₁ = (1, 0) and e₂ = (0, 1).

线性变换保持原点不动，并把直线映射为直线。这意味着经过矩阵变换后，线段的像仍是线段，平行线仍保持平行。只要知道变换对两个单位向量 e₁ = (1, 0) 和 e₂ = (0, 1) 的作用，整个变换就被完全确定。

The columns of M are exactly the images of e₁ and e₂. The first column tells you where (1, 0) goes, and the second column tells you where (0, 1) goes. This is the single most useful idea for identifying a transformation from its matrix, and it is the quickest way to check an answer in the exam.

矩阵 M 的两列正好就是 e₁ 和 e₂ 的像。第一列告诉你 (1, 0) 去了哪里，第二列告诉你 (0, 1) 去了哪里。这是从矩阵反推变换类型时最有用的思路，也是考试中检查答案最快的方法。

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2. The Identity Matrix | 单位矩阵

aves every point unchanged. For any vector v, Iv = v. In transformation terms, this is the 'do nothing' map: every point is fixed, and every line is invariant.

单位矩阵 I = [1 0; 0 1] 使每个点保持不变。对任意向量 v，都有 Iv = v。从变换角度看，这是"不做任何事"的映射：每个点都是不变点，每条直线都是不变直线。

matrix M. Its determinant is 1, so it has an area scale factor of 1, which matches the fact that areas do not change.

它是矩阵乘法的单位元：对于任意 2 × 2 矩阵 M，都有 MI = IM = M。它的行列式为 1，因此面积缩放因子为 1，这与面积不变的事实一致。

M I = I M = M,   det I = 1

pot an identity matrix inside a product, you can often simplify an expression immediately.

单位矩阵也出现在逆矩阵的定义中：M M⁻¹ = M⁻¹ M = I。如果你能在一个乘积中识别出单位矩阵，往往可以立刻简化表达式。

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3. Reflections | 反射变换

Reflections are represented by matrices with determinant −1. The determinant being negative shows that the transformation reverses orientation, which is exactly what a reflection does to the plane. The most common AQA reflection matrices are listed below.

反射由行列式为 −1 的矩阵表示。行列式为负说明该变换改变了平面的定向，而这正是反射对平面所做的事。下面列出 AQA 考试中最常见的反射矩阵。

Transformation 变换	Matrix 矩阵
Reflection in x-axis 关于 x 轴反射	[1 0; 0 −1]
Reflection in y-axis 关于 y 轴反射	[−1 0; 0 1]
Reflection in y = x 关于 y = x 反射	[0 1; 1 0]
Reflection in y = −x 关于 y = −x 反射	[0 −1; −1 0]

iant under the matrix [0 1; 1 0] because points on that line are swapped, so the line as a whole is mapped onto itself.

例如，矩阵 [1 0; 0 −1] 将 (x, y) 映射为 (x, −y)，即关于 x 轴的反射。直线 y = x 在矩阵 [0 1; 1 0] 下是不变的，因为该直线上的点被交换，整条直线被映射到自身。

matrix R, and it is a useful check when you are asked to find the inverse of a reflection.

反射是自身的逆变换：连续做两次同样的反射，每个点都会回到原位。这就是为什么任意反射矩阵 R 都满足 R² = I，这也是求反射逆矩阵时一个很有用的检验方法。

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4. Rotations | 旋转变换

os θ]. A clockwise rotation through angle θ is represented by [cos θ sin θ; −sin θ cos θ], which is the same as anticlockwise rotation through −θ.

绕原点逆时针旋转角度 θ 的矩阵为 Rθ = [cos θ −sin θ; sin θ cos θ]。顺时针旋转角度 θ 的矩阵为 [cos θ sin θ; −sin θ cos θ]，它等同于逆时针旋转 −θ。

Anticlockwise θ: [cos θ −sin θ; sin θ cos θ]    Clockwise θ: [cos θ sin θ; −sin θ cos θ]

erve area and do not reverse orientation. The special angles you must know are 90°, 180° and 270°, and it is worth memorising their matrices rather than deriving them each time.

旋转矩阵的行列式为 cos²θ + sin²θ = 1，因此旋转保持面积不变，也不改变定向。必须掌握的特殊角是 90°、180° 和 270°，建议直接记住它们的矩阵，而不是每次都重新推导。

Rotation 旋转	Matrix 矩阵
90° anticlockwise 逆时针 90°	[0 −1; 1 0]
180°	[−1 0; 0 −1]
90° clockwise 顺时针 90°	[0 1; −1 0]

0°, every line through the origin is invariant as a whole, because each point is mapped to the opposite side of the origin while remaining on the same line.

对于旋转，除 θ = 180° 外，原点通常是唯一的不变点。当 θ = 180° 时，每条过原点的直线整体上都是不变直线，因为每个点被映射到原点另一侧，但仍留在同一直线上。

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5. Enlargements and Stretches | 放大与伸缩

An enlargement of scale factor k with centre at the origin is represented by [k 0; 0 k]. Its determinant is k², so the area scale factor is k². If k is negative, the transformation is an enlargement combined with a half-turn about the origin.

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以原点为中心、比例因子为 k 的放大由 [k 0; 0 k] 表示。其行列式为 k²，因此面积缩放因子为 k²。如果 k 为负数，则该变换是放大与绕原点 180° 旋转的复合。

A one-way stretch is represented by [k 0; 0 1] (stretch parallel to the x-axis with scale factor k) or [1 0; 0 k] (stretch parallel to the y-axis with scale factor k). These have determinant k, so the area scale factor is |k|.

单向伸缩由 [k 0; 0 1]（平行于 x 轴、比例因子为 k 的伸缩）或 [1 0; 0 k]（平行于 y 轴、比例因子为 k 的伸缩）表示。它们的行列式为 k，因此面积缩放因子为 |k|。

Enlargement: [k 0; 0 k], det = k²    Stretch: [k 0; 0 1], det = k

llapses the whole plane onto a single line or point, which is why it has no inverse. Whenever you see a zero row or a zero column, you should immediately suspect that the determinant is zero.

当 k = 0 时矩阵是奇异的，会把整个平面压缩为一条直线或一个点，因此没有逆矩阵。每当你看到某一行或某一列为零时，就应立即怀疑行列式为零。

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6. Shears | 剪切变换

An x-direction shear is represented by [1 k; 0 1] and maps (x, y) to (x + ky, y). The y-coordinate is unchanged while the x-coordinate shifts by an amount proportional to y. Points on the x-axis do not move at all, so the x-axis is a line of fixed points.

x 方向剪切由 [1 k; 0 1] 表示，将 (x, y) 映射为 (x + ky, y)。y 坐标保持不变，而 x 坐标移动的量与 y 成正比。x 轴上的点完全不动，因此 x 轴是固定点组成的直线。

A y-direction shear is represented by [1 0; k 1] and maps (x, y) to (x, kx + y). Here the x-coordinate is unchanged and the y-axis is the line of fixed points. Both shear matrices have determinant 1, so shears preserve area even though they change the shape of a figure.

y 方向剪切由 [1 0; k 1] 表示，将 (x, y) 映射为 (x, kx + y)。此时 x 坐标保持不变，y 轴是固定点组成的直线。两种剪切矩阵的行列式都是 1，因此剪切虽然改变图形形状，却保持面积不变。

x-shear: [1 k; 0 1]    y-shear: [1 0; k 1]    det = 1

ch is still on y = c. This is a classic exam question about invariant lines that are not lines of fixed points.

对于剪切 [1 k; 0 1]，每条平行于 x 轴的直线整体上都是不变的：直线 y = c 被映射到自身，因为 (x, c) 变为 (x + kc, c)，仍在 y = c 上。这是一个关于"不变直线不一定是固定点直线"的经典考题。

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7. Determinants and Area Scale Factor | 行列式与面积缩放因子

nt gives the area scale factor of the transformation: a region of area A is mapped to a region of area |det M| , because a negative determinant means orientation is reversed but area is still positive.

对于 2 × 2 矩阵 M = [a b; c d]，其行列式为 det M = ad − bc。行列式给出了变换的面积缩放因子：面积为 A 的区域被映射为面积为 |det M| × A 的区域。绝对值必不可少，因为负的行列式只表示定向反转，面积仍然是正的。

det M = ad − bc    Area scale factor = |det M|

A useful way to see this is to look at the unit square with vertices (0, 0), (1, 0), (1, 1) and (0, 1). Its image is a parallelogram whose area is exactly |det M|. The columns of M are two sides of this parallelogram, so the determinant measures how much the transformation stretches or squashes area.

理解这一点的一个好方法是观察顶点为 (0, 0)、(1, 0)、(1, 1)、(0, 1) 的单位正方形。它的像是一个平行四边形，面积恰好为 |det M|。矩阵 M 的两列就是这个平行四边形的两条边，因此行列式衡量的是变换对面积的拉伸或压缩程度。

sts. If det M = 1, the transformation is area-preserving, as in the case of rotations and shears. If det M = −1, area is preserved but orientation is reversed, as in reflections.

如果 det M = 0，矩阵是奇异的。该变换把平面压缩为一条直线或一个点，面积缩放因子为 0，且不存在逆矩阵。如果 det M = 1，变换保持面积不变，旋转和剪切就是这种情况。如果 det M = −1，面积保持不变但定向反转，反射就是这种情况。

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8. Combining Transformations | 变换的复合

When two transformations are applied one after the other, the combined matrix is the product of the individual matrices. Order matters: if transformation A is applied first and transformation B second, the combined matrix is BA, not AB. The matrix closest to the vector acts first.

当两个变换依次作用时，复合矩阵就是各矩阵的乘积。顺序非常重要：如果先作用变换 A，再作用变换 B，那么复合矩阵是 BA，而不是 AB。最靠近列向量的那个矩阵最先作用。

Apply A then B: v → Av → B(Av) = (BA)v

Matrix multiplication is not commutative in general, so AB ≠ BA. A good example is a reflection followed by a rotation: rotating after reflecting usually gives a different result from reflecting after rotating. If a question asks for the single matrix representing two transformations, always write the matrices in the order in which they are applied, from right to left.

矩阵乘法一般不可交换，因此 AB ≠ BA。一个很好的例子是"先反射再旋转"与"先旋转再反射"：两者结果通常不同。如果题目要求用单个矩阵表示两个变换，一定要按作用顺序从右到左写出各矩阵。

You can always check a composite matrix by tracking the unit vectors. Find the image of (1, 0) and the image of (0, 1) step by step, and these two image vectors become the columns of the composite matrix.

你总可以通过追踪单位向量来检验复合矩阵。逐步求出 (1, 0) 的像和 (0, 1) 的像，这两个像向量就构成复合矩阵的两列。

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9. Inverse Matrices and Inverse Transformations | 逆矩阵与逆变换

rse matrix undoes the original transformation: applying M and then M⁻¹ returns every point to its starting position.

如果 det M ≠ 0，则 M = [a b; c d] 的逆矩阵为 M⁻¹ = 1/(ad − bc) [d −b; −c a]。逆矩阵抵消原变换的作用：先作用 M 再作用 M⁻¹，每个点都会回到起始位置。

M⁻¹ = 1/(ad − bc) [d −b; −c a]    M M⁻¹ = M⁻¹ M = I

The inverse is used to solve matrix equations. If Mv = b, then multiplying both sides

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