📚 Linear Transformations | 线性变换
A linear transformation is a core concept in A-Level mathematics. It maps points in a plane (or space) to new points using a matrix, while keeping straight lines straight and the origin fixed. Understanding how matrices encode transformations is essential for solving problems about reflections, rotations, enlargements and combinations of these operations.
线性变换是 A-Level 数学的核心概念。它通过矩阵将平面（或空间）中的点映射到新位置，同时保持直线仍然是直线、原点保持不动。理解矩阵如何编码变换，对于解决反射、旋转、放大以及这些操作的复合问题至关重要。
1. What is a Linear Transformation? | 什么是线性变换？
A transformation is linear if it satisfies two algebraic rules. First, adding two vectors before transforming gives the same result as transforming them separately and then adding: T(u + v) = T(u) + T(v). Second, scaling a vector before transforming is the same as transforming and then scaling: T(ku) = kT(u). These two rules ensure that the transformation is completely determined by its action on basis vectors.
如果一个变换满足两条代数规则，它就是线性的。第一，先对两个向量相加再变换，与分别变换后再相加的结果相同：T(u + v) = T(u) + T(v)。第二，先对向量进行数乘再变换，与先变换再进行数乘的结果相同：T(ku) = kT(u)。这两条规则保证了变换完全由其在基向量上的作用决定。
In the 2D plane, we use the standard basis vectors i = (1, 0) and j = (0, 1). If we know where i and j go, we can build the matrix for the transformation. The image of i becomes the first column, and the image of j becomes the second column.
在二维平面中，我们使用标准基向量 i = (1, 0) 和 j = (0, 1)。如果我们知道 i 和 j 变换到哪里，就可以构造出该变换的矩阵：i 的像成为第一列，j 的像成为第二列。
2. Matrix Representation | 矩阵表示
In 2D, a linear transformation is represented by a 2 x 2 matrix M = [a b; c d]. When we apply M to a position vector (x, y), we multiply: M (x y)ᵀ = (a x + b y, c x + d y)ᵀ. The transformed coordinates are linear combinations of the original coordinates.
在二维中，线性变换由 2×2 矩阵 M = [a b; c d] 表示。当我们将 M 作用于位置向量 (x, y) 时，进行乘法：M (x y)ᵀ = (a x + b y, c x + d y)ᵀ。变换后的坐标是原坐标的线性组合。
|a b| |x| = |ax + by|
|c d| |y| |cx + dy|
We conventionally write the image of i as column 1 and the image of j as column 2. For example, if i maps to (1, 0) and j maps to (0, -1), the matrix is [1 0; 0 -1], which is a reflection in the x-axis.
我们约定将 i 的像写作第 1 列，将 j 的像写作第 2 列。例如，若 i 映射到 (1, 0)，j 映射到 (0, -1)，则矩阵为 [1 0; 0 -1]，这表示关于 x 轴的反射。
3. Unit Vectors and Column Construction | 单位向量与矩阵构造
To find the matrix of a transformation, apply it to i = (1, 0) and j = (0, 1). The result of T(i) is the first column, and T(j) is the second column. This method works for reflections, rotations, shears and enlargements.
要求一个变换的矩阵，可以将变换作用于 i = (1, 0) 和 j = (0, 1)。T(i) 的结果是第一列，T(j) 的结果是第二列。这种方法适用于反射、旋转、剪切和放大等变换。
Example: a rotation by 90 degrees counterclockwise sends i to (0, 1) and j to (-1, 0). Therefore the matrix is [0 -1; 1 0]. Using this column rule avoids memorising many separate formulas.
例如：逆时针旋转 90° 将 i 送到 (0, 1)，将 j 送到 (-1, 0)。因此矩阵为 [0 -1; 1 0]。使用这一“列规则”可以避免记忆许多独立的公式。
4. Reflections | 反射变换
Reflections in the x-axis, y-axis, the line y = x, and the line y = -x are common in exams. Reflection in the x-axis leaves i unchanged and sends j to (0, -1), so the matrix is [1 0; 0 -1]. Reflection in the line y = x swaps i and j, giving [0 1; 1 0].
关于 x 轴、y 轴、直线 y = x 和 y = -x 的反射在考试中很常见。关于 x 轴的反射保持 i 不变，将 j 送到 (0, -1)，因此矩阵为 [1 0; 0 -1]。关于直线 y = x 的反射交换 i 和 j，得到 [0 1; 1 0]。
Reflection in an arbitrary line through the origin, making an angle θ with the x-axis, has a more general matrix. However, in A-Level questions you are usually asked either to construct the matrix from the images of i and j, or to identify the transformation from a given matrix.
关于过原点且与 x 轴成 θ 角的任意直线的反射有一个更一般的矩阵。但在 A-Level 题目中，通常要求你根据 i 和 j 的像构造矩阵，或者根据给定矩阵识别变换类型。
5. Rotations | 旋转变换
For a counterclockwise rotation by angle θ about the origin, the matrix is [cos θ -sin θ; sin θ cos θ]. This is derived by rotating i to (cos θ, sin θ) and j to (-sin θ, cos θ). If the rotation is clockwise, we use -θ, and the matrix becomes [cos θ sin θ; -sin θ cos θ].
绕原点逆时针旋转 θ 角的矩阵为 [cos θ -sin θ; sin θ cos θ]。这是通过将 i 旋转到 (cos θ, sin θ)，将 j 旋转到 (-sin θ, cos θ) 推导出来的。如果旋转方向为顺时针，则使用 -θ，矩阵变为 [cos θ sin θ; -sin θ cos θ]。
Special angles like 90°, 180° and 270° appear frequently. For 180°, cos 180° = -1 and sin 180° = 0, so the matrix is [-1 0; 0 -1], which is a half-turn. This is also equivalent to a scaling by factor -1 in both directions.
90°、180° 和 270° 等特殊角经常出现。对于 180°，cos 180° = -1，sin 180° = 0，因此矩阵为 [-1 0; 0 -1]，即半圈旋转。它也等价于两个方向都乘以 -1 的缩放。
6. Scaling and Shearing | 缩放与剪切
A uniform scaling by factor k has matrix [k 0; 0 k]. A non-uniform scaling scales x by a and y by b, giving [a 0; 0 b]. For example, [2 0; 0 3] stretches the x-direction by 2 and the y-direction by 3.
比例因子为 k 的均匀缩放矩阵为 [k 0; 0 k]。非均匀缩放将 x 方向缩放 a、y 方向缩放 b，得到 [a 0; 0 b]。例如，[2 0; 0 3] 将 x 方向拉伸 2 倍，将 y 方向拉伸 3 倍。
A shear in the x-direction has the form [1 k; 0 1]. It moves points horizontally by an amount proportional to their y-coordinate, while leaving the y-coordinate unchanged. The image of i is still (1, 0), and the image of j is (k, 1). Shears preserve area, which is reflected in the determinant being 1.
x 方向剪切的矩阵形式为 [1 k; 0 1]。它使点水平移动的量与其 y 坐标成正比，同时保持 y 坐标不变。i 的像仍是 (1, 0)，j 的像是 (k, 1)。剪切保持面积不变，这体现在行列式为 1 上。
7. Combined Transformations | 复合变换
When two linear transformations A and B are applied successively, the resulting single transformation is their matrix product. Careful: if B is applied first and then A, the overall matrix is AB. Matrix multiplication is not commutative, so AB is generally not equal to BA.
当两个线性变换 A 和 B 依次作用时，合成的单一变换是它们的矩阵乘积。注意：如果先施加 B，再施加 A，则总矩阵是 AB。矩阵乘法不满足交换律，所以 AB 通常不等于 BA。
Example: reflect in the y-axis using matrix R = [-1 0; 0 1], then rotate by 90° counterclockwise using matrix Q = [0 -1; 1 0]. The combined matrix is QR = [0 -1; 1 0] × [-1 0; 0 1] = [0 -1; -1 0]. This shows that the order of operations matters for the final image.
示例：先用矩阵 R = [-1 0; 0 1] 关于 y 轴反射，然后用矩阵 Q = [0 -1; 1 0] 逆时针旋转 90°。复合矩阵为 QR = [0 -1; 1 0] × [-1 0; 0 1] = [0 -1; -1 0]。这表明操作的顺序会影响最终图像。
In exam questions, always check which transformation is applied ‘first’. The matrix written on the right of the product is the first applied transformation. Incorrect order is a common source of lost marks.
在考试题目中，始终要检查哪个变换是“先”施加的。乘积中写在右侧的矩阵是首先施加的变换。顺序错误是常见的失分点。
8. Inverse Transformations | 逆变换
The inverse of a matrix M undoes the effect of M. For a 2 x 2 matrix M = [a b; c d], the inverse is (1/det(M)) × [d -b; -c a], provided the determinant det(M) = ad – bc is not zero. If det(M) = 0, the transformation collapses the plane onto a line or a point, and no inverse exists.
矩阵 M 的逆矩阵能够撤销 M 的效果。对于 2×2 矩阵 M = [a b; c d]，其逆矩阵为 (1/det(M)) × [d -b; -c a]，前提是行列式 det(M) = ad – bc 不等于零。如果 det(M) = 0，变换将平面压缩到一条线或一个点，此时不存在逆变换。
Geometrically, the inverse of a counterclockwise rotation by θ is a clockwise rotation by θ. The inverse of a reflection is the same reflection, since applying it twice returns the original shape. Recognising these geometric inverses can save time.
从几何上看，逆时针旋转 θ 的逆变换就是顺时针旋转 θ。反射的逆变换是它自身，因为施加两次就回到原始图形。识别这些几何逆变换可以节省时间。
9. Determinant and Area Scale Factor | 行列式与面积缩放因子
The determinant of the matrix that represents a linear transformation gives the factor by which the area of any shape is multiplied. A 2 x 2 matrix with determinant d changes an area of A into an area of |d|A. The absolute value is used because area is always positive.
表示线性变换的矩阵的行列式给出了任意图形面积被乘的因子。一个 2×2 矩阵，若其行列式为 d，则会把面积 A 变成 |d|A。使用绝对值是因为面积总是正的。
If det(M) = 2, every shape doubles in area. If det(M) = -3, the area is tripled and the orientation is reversed. If det(M) = 0, the transformation is singular and maps the whole plane onto a lower-dimensional set, so the area becomes zero.
如果 det(M) = 2，任何图形面积变为原来的 2 倍。如果 det(M) = -3，面积变为 3 倍并且方向被翻转。如果 det(M) = 0，变换是奇异的，将整个平面映射到低维集合上，所以面积变为零。
For a product of matrices AB, we have det(AB) = det(A)det(B). This is useful when a transformation is built from multiple steps: the area scale factor of the combined transformation is the product of the individual scale factors.
对于矩阵乘积 AB，有 det(AB) = det(A)det(B)。当变换由多个步骤构成时，这一性质很有用：复合变换的面积缩放因子是各步缩放因子的乘积。
10. Extension to 3D and Exam Tips | 三维扩展与考试提示
Linear transformations also work in 3D using 3 x 3 matrices. The images of the standard vectors i, j and k become the three columns. The determinant then represents the volume scale factor. For example, a reflection in the xy-plane has matrix [1 0 0; 0 1 0; 0 0 -1].
线性变换同样适用于三维，使用 3×3 矩阵。标准向量 i、j 和 k 的像成为三个列。此时行列式表示体积缩放因子。例如，关于 xy 平面的反射矩阵为 [1 0 0; 0 1 0; 0 0 -1]。
For AQA A-Level mathematics, key revision points are: writing a transformation matrix from unit vector images, recognising standard reflections and rotations, multiplying matrices for composite transformations, calculating the inverse using the determinant, and using the determinant for area and volume scale factors. Practise these with past paper questions.
对于 AQA A-Level 数学，关键复习要点包括：根据单位向量的像写出变换矩阵，识别标准反射和旋转，用矩阵乘法处理复合变换，利用行列式计算逆矩阵，以及用行列式求面积和体积缩放因子。请结合往届真题进行练习。
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