Linear transformations

In Lesson 13 a matrix was a grid of numbers with a rule for multiplying. This lesson tells you what those numbers mean. A linear transformation is a function that takes the whole plane and re-draws it - but only in the tame, disciplined ways where grid lines stay straight, stay parallel, and stay evenly spaced, and the origin never budges. Every such re-drawing of 2-D space, no matter how twisted it looks, is captured by just four numbers: a $2\times 2$ matrix. Drag two arrows below and watch space itself bend to follow.

Linear = grid lines stay straight and evenly spaced

Out of all the ways you could distort a rubber sheet, transformations earn the name linear when they obey two rules. First, they respect addition - transforming a sum is the same as transforming the parts and adding. Second, they respect scaling - stretching a vector then transforming equals transforming then stretching:

Geometrically these two algebraic rules cash out as a picture you can check by eye: lines stay lines (no curving), parallel lines stay parallel, evenly spaced tick marks stay evenly spaced, and - because $T(0)=T(0\cdot \mathbf{v})=0\cdot T(\mathbf{v})=0$ - the origin is nailed in place. That last fact is why sliding the whole plane sideways (a translation) is not linear: it drags the origin along with it.

The matrix is just where $\hat{ı}$ and $\hat{ȷ}$ land

Here is the trick that makes those four numbers appear. Any vector is built from the two basis vectors $\hat{ı}=(1,0)$ and $\hat{ȷ}=(0,1)$ by scaling and adding: $\mathbf{v}=x\hat{ı}+y\hat{ȷ}$. Feed that through a linear map and the two rules let you pull the transformation inside the sum:

So the output is completely determined by the two landing spots $T(\hat{ı})$ and $T(\hat{ȷ})$. Stack them as the columns of a matrix and you get exactly the machine from Lesson 13:

Four numbers pin down the fate of an infinite plane. In the demo, the two arrows you drag are the two columns: move $\hat{ı}$'s tip and you rewrite the left column; move $\hat{ȷ}$'s tip and you rewrite the right one.

A small zoo of transformations

Every preset button above is a famous matrix. Once you read matrices as "where the basis lands," each one is memorable rather than mysterious:

Rotation and shear keep every area intact ($\det =1$); scaling multiplies area by $s_x{s}_y$; reflection preserves area but reverses handedness ($\det =-1$). To do one transformation then another, you compose them - and composition is exactly matrix multiplication: applying $B$ then $A$ is the single matrix $AB$. This is why Lesson 13 defined matmul the strange way it did, and why order matters ($AB\ne BA$): spinning then stretching is not the same as stretching then spinning.

import numpy as np

# columns are where î and ĵ land -- that IS the matrix
i_hat = np.array([1, 0])          # î stays put
j_hat = np.array([1, 1])          # ĵ slides right -> a shear
M = np.column_stack([i_hat, j_hat])   # [[1, 1], [0, 1]]

v  = np.array([2, 1])
Mv = M @ v                       # = 2*i_hat + 1*j_hat = [3, 1]

np.linalg.det(M)                 # 1.0  -> area unchanged, orientation kept

R = np.array([[-1, 0], [0, 1]])  # reflect across the y-axis
np.linalg.det(R)                 # -1.0 -> orientation flipped, space inside-out

Check yourself