Linear Transformations and Matrices
Введено| English | Значение |
|---|---|
| linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/ | линейное преобразование |
| unit square/ˈjuːnɪt skweə/ | единичный квадрат |
| shear/ʃɪə/ | сдвиг |
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.
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.
A linear transformation of the plane always…
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
A linear transformation keeps the origin fixed and maps straight, evenly-spaced grid lines to straight, evenly-spaced lines.
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.
The columns of the matrix tell you where the two basis ____ land.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Column 1 is where $\langle1,0\rangle$ lands and column 2 is where $\langle0,1\rangle$ lands — that fully describes the transformation.
The matrix $\begin{bmatrix}2&1\\0&1\end{bmatrix}$ sends $\langle1,1\rangle$ до $\langle 2\cdot1+1\cdot1,\ 1\rangle$. What is the new $x$?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Row 1 dotted with $\langle1,1\rangle$: $2\cdot1 + 1\cdot1 = 3$, so the image is $\langle3, 1\rangle$.
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.
Linear transformations as matrices
A rotation is a linear transformation; this matrix turns the plane a quarter turn.
Doing one linear transformation after another corresponds to adding their matrices.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
False. Composition of linear transformations corresponds to matrix multiplication, which is why the order of the product matters.
Select все true statements about linear transformations.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Linear transformations keep lines straight — they never curve the grid.
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.
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.
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.
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.
Write from an image x-coordinate of 6
- The matrix with rows (3, 0) and (0, 1) sends the point (2, 5).
- Find the x-coordinate of the image.
The matrix with rows (3, 0) and (0, 1) is applied to the point (2, 5). What is the x-coordinate of the image?
3 times 2 is 6.
What is the image x-coordinate?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The first row scales x by 3. Other options — 3 (the lesson image coordinate); 5 (the original y-coordinate).