Linear Transformations and Matrices · 线性变换与矩阵
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/ | 线性变换 | xiàn xìng biàn huàn |
| unit square/ˈjuːnɪt skweə/ | 单位正方形 | dān wèi zhèng fāng xíng |
| shear/ʃɪə/ | 错切 | cuò qiè |
A matrix moves points
- A matrix does more than store numbers — it can move every point of the plane.
- Feed it a vector, and it hands back a new, transformed vector.
- Squares can become slanted; grids can stretch or rotate.
- This turns a matrix into a machine for reshaping space.
矩阵移动点
- 矩阵不只是储存数字——它能移动平面上的每一个点。
- 给它一个向量,它就交回一个新的、被变换过的向量。
- 正方形可以变成倾斜的;网格可以拉伸或旋转。
- 这让矩阵成为一台重塑空间的机器。
What a linear transformation is
- A linear transformation 线性变换 is a function that maps every vector to a new vector.
- It keeps the origin fixed and keeps grid lines straight and evenly spaced.
- No curving, no tearing — parallel lines stay parallel.
- Every linear transformation of the plane can be written as multiplication by a matrix.
什么是线性变换
- 线性变换(linear transformation)是一个把每个向量映射到新向量的函数。
- 它保持原点不动,并让网格线保持直且等距。
- 不弯曲、不撕裂——平行线保持平行。
- 平面的每一个线性变换都能写成乘以一个矩阵。
A linear transformation of the plane always… · 平面的线性变换总是…
A linear transformation keeps the origin fixed and maps straight, evenly-spaced grid lines to straight, evenly-spaced lines. · 一个线性变换保持原点固定，并将直线、等距的网格线映射为直线、等距的线。
The columns say where the axes go
- Column 1 shows where the vector $\langle 1, 0\rangle$ lands; column 2, where $\langle 0, 1\rangle$ lands.
- Knowing those two images tells you where every point goes.
- To transform a vector, multiply the matrix by it (row-by-column).
- The unit square 单位正方形 becomes a parallelogram — its shape reveals the transformation.
列告诉你坐标轴去哪
- 第 1 列显示向量 $\langle 1, 0\rangle$ 落在哪里;第 2 列,$\langle 0, 1\rangle$ 落在哪里。
- 知道这两个像,就知道每一个点去哪。
- 要变换一个向量,把矩阵乘以它(行乘列)。
- 单位正方形(unit square)变成一个平行四边形——它的形状揭示了变换。
The columns of the matrix tell you where the two basis ____ land. · 矩阵的列告诉你两个基向量____落在哪里。
Column 1 is where $\langle1,0\rangle$ lands and column 2 is where $\langle0,1\rangle$ lands — that fully describes the transformation. · 第1列是$\langle1,0\rangle$落下的位置，第2列是$\langle0,1\rangle$落下的位置——这完全描述了该变换。
The matrix $\begin{bmatrix}2&1\\0&1\end{bmatrix}$ sends $\langle1,1\rangle$ to · 到 $\langle 2\cdot1+1\cdot1,\ 1\rangle$. What is the new $x$? · 矩阵$\begin{bmatrix}2&1\\0&1\end{bmatrix}$将$\langle1,1\rangle$发送到$\langle 2\cdot1+1\cdot1,\ 1\rangle$。新的$x$是什么？
Row 1 dotted with $\langle1,1\rangle$: $2\cdot1 + 1\cdot1 = 3$, so the image is $\langle3, 1\rangle$. · 第1行与$\langle1,1\rangle$点乘：得到$2\cdot1 + 1\cdot1 = 3$，因此图像是$\langle3, 1\rangle$。
Familiar transformations
- Scaling: a diagonal matrix stretches or shrinks along the axes.
- Rotation: a special matrix spins the plane around the origin.
- Shear 错切: slides one direction while holding the other — like italic text.
- Reflection: flips the plane across a line.
熟悉的变换
- 缩放:一个对角矩阵沿坐标轴拉伸或收缩。
- 旋转:一个特殊矩阵让平面绕原点旋转。
- 错切(shear):在保持一个方向不动的同时滑动另一个方向——像斜体字。
- 反射:把平面沿一条线翻转。
Linear transformations as matrices · 作为矩阵的线性变换
A rotation is a linear transformation; this matrix turns the plane a quarter turn. · 旋转是一种线性变换；该矩阵将平面旋转四分之一圈。
Doing one linear transformation after another corresponds to multiplying their matrices. · 连续执行两次线性变换相当于对它们的矩阵进行乘法运算。
Composition of transformations is matrix multiplication — that is why the product order matters. · 变换的复合就是矩阵乘法——这就是为什么乘积的顺序很重要。
Select all · 所有 true statements about linear transformations. · 选择关于线性变换的所有正确陈述。
Linear transformations keep lines straight — they never curve the grid. · 线性变换保持线条笔直——它们绝不会使网格弯曲。
Combining transformations
- Do one transformation, then another: the result is a single new transformation.
- Its matrix is the product of the two matrices.
- Because matrix products depend on order, "rotate then shear" differs from "shear then rotate".
- This is why computer graphics stack transformations by multiplying matrices.
组合变换
- 先做一个变换,再做另一个:结果是一个单一的新变换。
- 它的矩阵是两个矩阵的乘积。
- 因为矩阵乘积依赖顺序,"先转再切"不同于"先切再转"。
- 这就是计算机图形学通过矩阵相乘来叠加变换的原因。
The order of combined transformations matters. Applying $A$ then $B$ is the matrix product $BA$ — the second transformation's matrix goes on the left. Swapping the order usually gives a different result, just as $BA \ne AB$ for matrices.
组合变换的顺序很重要。先做 $A$ 再做 $B$,是矩阵乘积 $BA$——第二个变换的矩阵放在左边。交换顺序通常给出不同的结果,正如对矩阵有 $BA \ne AB$。
Apply $\begin{bmatrix}2&1\\0&1\end{bmatrix}$ to the corners of the unit square.
- $\langle1,0\rangle \to \langle2,0\rangle$ — the bottom edge doubles in length.
- $\langle0,1\rangle \to \langle1,1\rangle$ — the top slides right (a shear).
- The square becomes a slanted parallelogram, exactly as the figure shows.
把 $\begin{bmatrix}2&1\\0&1\end{bmatrix}$ 作用到单位正方形的角上。
- $\langle1,0\rangle \to \langle2,0\rangle$——底边长度翻倍。
- $\langle0,1\rangle \to \langle1,1\rangle$——顶部向右滑动(一个错切)。
- 正方形变成一个倾斜的平行四边形,正如图中所示。
A linear transformation maps every vector to a new one, keeping the origin fixed and grid lines straight — and every one is multiplication by a matrix. The matrix's columns are where the axes land, so it turns the unit square into a parallelogram. Combining transformations means multiplying matrices, and order matters.
线性变换把每个向量映射到一个新向量,保持原点不动、网格线保持直——而且每一个都是乘以一个矩阵。矩阵的列是坐标轴落到的地方,所以它把单位正方形变成一个平行四边形。组合变换意味着矩阵相乘,而且顺序重要。