Linear Transformations and Matrices
| English | Chinese | Pinyin |
|---|---|---|
| linear transformation | 线性变换 | xiàn xìng biàn huàn |
| unit square | 单位正方形 | dān wèi zhèng fāng xíng |
| shear | 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.
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…
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.
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 $\langle1,1\rangle$ to $\langle 2\cdot1+1\cdot1,\ 1\rangle$. What is the new $x$?
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 multiplying their matrices.
Composition of transformations is matrix multiplication — that is why the product order matters.
Select all true statements about linear transformations.
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.