Matrix Transformations (2-dimensional) | 二维矩阵变换

📚 Matrix Transformations (2-dimensional) | 二维矩阵变换

Matrix transformations are one of the most visual and rewarding topics in A-Level Mathematics. A 2×2 matrix can be used to transform any point (x, y) in the plane to a new point (x’, y’), and by understanding how matrices act on vectors, you can describe reflections, rotations, enlargements and shears with precision and elegance.

矩阵变换是A-Level数学中最具直观性和趣味性的主题之一。一个2×2矩阵可以用来将平面上的任意点(x, y)变换到新点(x’, y’)，通过理解矩阵如何作用于向量，你可以精确而优雅地描述反射、旋转、放大和剪切等变换。

1. Representing Transformations with Matrices | 用矩阵表示变换

To transform a point (x, y), we write it as a column vector and multiply it by a 2×2 matrix M:

[x’] = [a b] [x]

[y’] [c d] [y]

This gives the equations x’ = ax + by and y’ = cx + dy. The first column of M tells us where the point (1, 0) — the unit vector in the x-direction — is mapped to, and the second column tells us where (0, 1) is mapped to. This is the key fact for finding the matrix of any transformation.

要将点(x, y)进行变换，我们将其写成列向量，并乘以一个2×2矩阵M：

[x’] = [a b] [x]

[y’] [c d] [y]

这给出方程x’ = ax + by 和 y’ = cx + dy。矩阵M的第一列告诉我们点(1, 0)（x方向的单位向量）被映射到何处，第二列告诉我们点(0, 1)被映射到何处。这是求任何变换矩阵的关键事实。

2. The Identity and Zero Matrices | 单位矩阵与零矩阵

The identity matrix I leaves every point unchanged:

I = [1 0; 0 1]

Multiplying any vector by I gives the same vector, just as multiplying a number by 1 leaves it unchanged. The zero matrix O maps every point to the origin (0, 0), collapsing the entire plane to a single point.

单位矩阵I使每个点保持不变：

I = [1 0; 0 1]

任何向量乘以I得到相同的向量，就像数乘以1不变一样。零矩阵O将每个点映射到原点(0, 0)，将整个平面压缩为一个点。

3. Reflection Transformations | 反射变换

A reflection flips points across a line through the origin. The four most common reflection matrices you must memorise for AQA are:

反射是绕通过原点的直线翻转点。AQA考试中必须牢记的四种最常见的反射矩阵如下：

Line of reflection	Matrix	Effect on (1,0),(0,1)
x-axis	[1 0; 0 −1]	(1,0)→(1,0), (0,1)→(0,−1)
y-axis	[−1 0; 0 1]	(1,0)→(−1,0), (0,1)→(0,1)
y = x	[0 1; 1 0]	(1,0)→(0,1), (0,1)→(1,0)
y = −x	[0 −1; −1 0]	(1,0)→(0,−1), (0,1)→(−1,0)

Notice that in every reflection, the determinant is −1. This reflects the fact that a reflection reverses orientation — clockwise becomes anticlockwise, and vice versa.

注意每次反射的行列式都是−1。这反映了反射改变方向的事实——顺时针变为逆时针，反之亦然。

4. Rotation Transformations | 旋转变换

To rotate a point by an angle θ anticlockwise about the origin, use the general rotation matrix:

R(θ) = [cos θ −sin θ; sin θ cos θ]

This is the single most important matrix in the entire topic. From it, you can derive all special cases. For example, a 90° anticlockwise rotation has cos 90° = 0 and sin 90° = 1, giving the matrix:

要将一个点绕原点逆时针旋转角度θ，使用一般旋转矩阵：

R(θ) = [cos θ −sin θ; sin θ cos θ]

这是整个主题中最重要的矩阵。从中可以推导出所有特殊情况。例如，逆时针旋转90°时cos 90° = 0，sin 90° = 1，得到矩阵：

Rotation	Matrix
90° anticlockwise	[0 −1; 1 0]
180°	[−1 0; 0 −1]
270° anticlockwise (or 90° clockwise)	[0 1; −1 0]

For clockwise rotations, substitute a negative angle: R(−θ) = [cos θ sin θ; −sin θ cos θ]. The determinant of every rotation matrix is 1, since cos²θ + sin²θ = 1, confirming that rotations preserve area and orientation.

对于顺时针旋转，代入负角度：R(−θ) = [cos θ sin θ; −sin θ cos θ]。每个旋转矩阵的行列式都是1，因为cos²θ + sin²θ = 1，这确认了旋转保持面积和方向不变。

5. Enlargement and Stretch Transformations | 放大与拉伸变换

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

[k 0; 0 k]

If k > 1 the shape grows; if 0 < k < 1 it shrinks; if k is negative the shape is also reflected through the origin. A stretch in the x-direction only multiplies the x-coordinate, keeping y unchanged:

以原点为中心、缩放因子为k的放大变换表示为：

[k 0; 0 k]

如果k > 1则图形变大；如果0 < k < 1则缩小；如果k为负数则图形还通过原点被反射。仅在x方向上的拉伸只乘以x坐标，保持y不变：

Stretch in x-direction: [k 0; 0 1]

Stretch in y-direction: [1 0; 0 k]

Note the difference between an enlargement and a stretch: an enlargement scales both coordinates by the same factor, preserving the shape; a stretch scales only one coordinate, distorting the shape in one direction.

注意放大和拉伸的区别：放大将两个坐标按相同因子缩放，保持形状不变；拉伸只缩放一个坐标，使形状沿一个方向变形。

6. Shear Transformations | 剪切变换

A shear in the x-direction fixes the y-coordinate and slides points horizontally by an amount proportional to their y-coordinate. Its matrix is:

[1 k; 0 1]

x方向的剪切固定y坐标，并使点沿水平方向滑动，滑动量与它们的y坐标成正比。其矩阵为：

[1 k; 0 1]

Applying this to point (x, y) gives (x + ky, y). The constant k is called the shear factor. Similarly, a shear in the y-direction has matrix:

[1 0; k 1]

将此矩阵作用于点(x, y)得到(x + ky, y)。常数k称为剪切因子。类似地，y方向的剪切矩阵为：

[1 0; k 1]

A shear keeps the area of a shape unchanged because its determinant is always 1. Points on the axis of the shear (the x-axis for an x-shear) remain fixed.

剪切保持图形面积不变，因为其行列式始终为1。在剪切轴上的点（x剪切时的x轴上的点）保持不变。

7. Combined Transformations | 复合变换

When two transformations are applied in sequence, the matrices are multiplied together — but the order matters critically. If transformation A is applied first, followed by transformation B, the combined matrix is BA, not AB:

Final = B(Av) = (BA)v

当两个变换按顺序作用时，矩阵要相乘——但顺序至关重要。如果先施加变换A，再施加变换B，则复合矩阵是BA，而不是AB：

最终结果 = B(Av) = (BA)v

For example, suppose we first rotate by 90° anticlockwise (matrix R) and then reflect in the x-axis (matrix F). The combined matrix is:

F R = [1 0; 0 −1] × [0 −1; 1 0] = [0 −1; −1 0]

which is the matrix for reflection in the line y = −x. If we reversed the order — reflect first, then rotate — we would get RF, which equals [0 1; 1 0], the reflection in y = x. This clearly demonstrates that multiplication of matrices is not commutative.

例如，假设我们先逆时针旋转90°（矩阵R），然后在x轴方向反射（矩阵F）。复合矩阵为：

F R = [1 0; 0 −1] × [0 −1; 1 0] = [0 −1; −1 0]

这是直线y = −x的反射矩阵。如果颠倒顺序——先反射再旋转——则得到RF，等于[0 1; 1 0]，即y = x的反射。这清楚地证明了矩阵乘法不满足交换律。

8. Inverse Transformations | 逆变换

For a 2×2 matrix M = [a b; c d], the inverse matrix M⁻¹ reverses the effect of M. The formula is:

M⁻¹ = 1/(ad − bc) × [d −b; −c a]

对于2×2矩阵M = [a b; c d]，逆矩阵M⁻¹使M的效果反转。公式为：

M⁻¹ = 1/(ad − bc) × [d −b; −c a]

The quantity ad − bc is the determinant, usually written |M| or det(M). If det(M) = 0, then M has no inverse; such a transformation maps the plane onto a line or a point, collapsing the area to zero, and cannot be reversed.

量ad − bc称为行列式，通常写作|M|或det(M)。如果det(M) = 0，则M没有逆矩阵；这样的变换将平面映射到一条线或一个点，将面积压缩为零，无法逆转。

Some inverse transformations are geometrically obvious. The inverse of a 90° anticlockwise rotation is a 90° clockwise rotation; the inverse of a reflection is the reflection itself, since reflecting twice in the same line returns every point to its original position.

有些逆变换在几何上显而易见。逆时针旋转90°的逆变换是顺时针旋转90°；反射的逆变换是反射本身，因为同一直线反射两次使每个点回到原来位置。

9. The Area Scaling Factor | 面积缩放因子

The absolute value of the determinant of a transformation matrix gives the area scale factor — the factor by which areas are multiplied when the transformation is applied to a shape.

变换矩阵行列式的绝对值给出面积缩放因子——即对图形施加变换时面积乘以的因子。

Area of transformed shape = |det(M)| × Area of original shape

For example, a square of area 4 transformed by M = [2 0; 0 3] becomes a rectangle of area 4 × |6| = 24. If det(M) = 0, the transformed shape has zero area and lies entirely on a straight line. If det(M) is negative, the shape is reflected and the area scale factor is still |det(M)|.

变换后图形的面积 = |det(M)| × 原图形的面积

例如，面积为4的正方形经M = [2 0; 0 3]变换后成为面积为4 × |6| = 24的矩形。如果det(M) = 0，变换后的图形面积为0，完全落在一条直线上。如果det(M)为负数，图形被反射，而面积缩放因子仍为|det(M)|。

10. Finding the Matrix from a Described Transformation | 由变换描述求矩阵

To find the matrix of any linear transformation, simply apply the transformation to the two basis vectors (1, 0) and (0, 1), and let their images form the columns of the matrix. This technique is reliable and always works, even for unusual transformations.

要求任何线性变换的矩阵，只需将变换作用于两个基向量(1, 0)和(0, 1)，让它们的像构成矩阵的列。这种技巧可靠且始终有效，即使对不常见的变换也适用。

Worked example: A transformation reflects points in the line y = (tan 30°)x, i.e. the line making an angle of 30° with the x-axis. The general formula for reflection in a line through the origin at angle α is:

[cos 2α sin 2α; sin 2α −cos 2α]

Substituting α = 30° gives cos 60° = ½ and sin 60° = √3/2, so the matrix is:

[½ √3/2; √3/2 −½]

示例：一个变换在直线y = (tan 30°)x上反射点，即与x轴成30°角的直线。过原点与x轴成角α的直线的反射一般公式为：

[cos 2α sin 2α; sin 2α −cos 2α]

代入α = 30°得到cos 60° = ½，sin 60° = √3/2，因此矩阵为：

[½ √3/2; √3/2 −½]

11. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always determine the order of combined transformations: the transformation applied first is written on the right. “First A, then B” gives the matrix BA.

始终确定复合变换的顺序：先施加的变换写在右边。”先A后B”得到的矩阵是BA。
Check whether det ≠ 0 before attempting to find an inverse. If det = 0, the transformation is singular and no inverse exists.

在求逆矩阵之前检查det ≠ 0。如果det = 0，变换是奇异的，不存在逆矩阵。
Use the unit vector method to verify your matrix: multiply your matrix by (1, 0) and (0, 1) and check that the outputs match the described transformation.

用单位向量方法检验矩阵：将矩阵乘以(1, 0)和(0, 1)，检查输出是否与描述的变换匹配。
Remember that an anticlockwise rotation through θ has matrix [cos θ −sin θ; sin θ cos θ]. A common mistake is to swap the signs of sin θ. Test with θ = 90°: (1, 0) must go to (0, 1), not (0, −1).

记住逆时针旋转θ的矩阵是[cos θ −sin θ; sin θ cos θ]。常见错误是搞错sin θ的正负号。用θ = 90°检验：(1, 0)必须到(0, 1)，而不是(0, −1)。
When asked for the area scale factor, give the absolute value of the determinant — a negative determinant still gives a positive area scale factor.

当被问及面积缩放因子时，给出行列式的绝对值——负的行列式仍然给出正的面积缩放因子。
For reflections, the lines y = x and y = −x are frequently tested. Memorise their matrices and also verify them by checking that reflecting (1, 0) lands on the correct point.

对于反射，直线y = x和y = −x是经常考查的。记住它们的矩阵，并通过检查(1, 0)反射后落在正确位置来验证。

12. Summary and Revision Checklist | 总结与复习清单

By this point you should be able to: write down the standard matrices for reflections, rotations, stretches and shears; combine transformations by multiplying matrices in the correct order; find the inverse matrix and understand when it exists; use the determinant to find the area scale factor; and reverse-engineer a matrix when given its geometric description.

至此，你应该能够：写出反射、旋转、拉伸和剪切的标准矩阵；按正确顺序通过矩阵乘法复合变换；求逆矩阵并理解其存在条件；利用行列式求面积缩放因子；以及在给定几何描述时逆向推导矩阵。

Keep this checklist in mind when revising: reflection in y = −x ↔ [0 −1; −1 0]; rotation 90° ↔ [0 −1; 1 0]; det = 0 means singular and non-invertible; combined transformations “first A then B” ↔ BA; area scale factor = |det|.

复习时记住这张清单：y = −x反射 ↔ [0 −1; −1 0]；旋转90° ↔ [0 −1; 1 0]；det = 0意味着奇异且不可逆；”先A后B”的复合变换 ↔ BA；面积缩放因子 = |det|。

Finally, practice as many past-paper questions as possible. Matrix transformations reward active problem-solving far more than passive reading. Draw sketches, test basis vectors, and check your answers using the unit vector method every time.

最后，尽可能多做历年真题。矩阵变换更青睐主动解题，而非被动阅读。画草图、检验基向量，并用单位向量方法每次核查你的答案。

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Matrix Transformations (2-dimensional) | 二维矩阵变换