Studio
Drag the mathematics.
Every studio belongs to one chapter and asks only what that chapter asked. They are not demonstrations to watch — nothing moves until you move it, and the sentence under the stage answers whatever you have just done to it. Each one also sits inside its chapter's Practice tab, next to the written exercises.
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001
Four houses, two dials, one walk downhill
The whole of machine learning is already in four rows of a spreadsheet: a rule with adjustable numbers, a way of saying how wrong it is, and a way of getting less wrong. These five scenes are that argument in the chapter's own order, and the two dials you turn in the first scene are the two coordinates you walk across in the fourth.
Two dials · How wrong · The landscape · Which way is downhill
Chapter 001 — What Does It Mean for a Machine to Learn → -
005
From a row of numbers to an arrow
The chapter's claim is that a list of features is a place, and a place is an arrow. These four scenes follow that claim through the chapter in order: hold one arrow, build new ones out of two, let a row of dials vote on a price, and then watch what unequal units do to all of it.
One arrow · Add and scale · The weighted vote · Different scales
Chapter 005 — Numbers Become Vectors → -
006
What a few arrows can build
The chapter takes Chapter 005's two operations — stretch an arrow, add two arrows — and asks how far they go. These six scenes are that question in order: one knob and the line it reaches, two knobs and the recipe they carry out, the cloud of everything reachable, the collapse when the second arrow is only a copy, the same story in three dimensions, and finally the same house read in two different languages.
One knob · Two knobs · Everywhere you can reach · A copy is not a direction · Line, plane, space · A different language
Chapter 006 — Directions, Span and Basis → -
007
One number for how much two arrows agree
Pair, multiply, add — and out comes a single number that knows about angles. These three scenes are the three things that number is: a sign, a shadow, and an angle once both lengths have been divided out.
The sign · The shadow · Only the angle · But size still counts · A classifier score
Chapter 007 — Dot Product: Measuring Alignment → -
008
A stack of questions, asked at once
One neuron is a dot product. A layer is many of them against the same input, and stacking the weight vectors as rows turns that pile of dot products into a single multiplication with a shape rule you can check before you compute anything.
Rows are questions · The shape rule · Every example at once · The shape ladder
Chapter 008 — Matrices: The Spreadsheet of Mathematics → -
009
Where the basis lands
A matrix is not a table of numbers that happens to multiply. It is a record of where the two basis vectors were sent, and everything else follows from that.
The columns · The whole grid · Scaling · Shear · Rotation · Reflection
Chapter 009 — Matrix-Vector Multiplication as Transformation → -
010
Two maps in a row are one map
Do one transformation and then another, and the pair is a third transformation you could have written down in advance. That third matrix is the product — and the reason the product depends on which one you do first is visible rather than algebraic.
First, then second · The other order · Undoing it · When there is no undo
Chapter 010 — Matrix-Matrix Multiplication as Composition → -
011
How much area survives
The determinant is not a formula to memorise. It is the factor by which a transformation multiplies area, and the moment it hits zero something is lost that no later step can recover.
Area factor · Turning over · Collapse · Almost collapsed
Chapter 011 — A Matrix Can Transform Space → -
012
What survives, and what a neuron does
Rank is how many directions come through a matrix alive, and the null space is what does not. With that in hand, a neuron stops being a cartoon: it is one learned direction and a threshold — and two of them stacked without a fold are still just one.
What vanishes · One neuron · Two layers, no fold · Nearly dependent
Chapter 012 — What a Matrix Does, and What a Neuron Is → -
014
The directions a transformation leaves alone
Try every direction and watch where the matrix sends it. Almost all of them swing away; one or two come back along the same line, only longer or shorter. Those are the eigenvectors — and some matrices have none at all.
Try every direction · When there is none · Again and again · Already diagonal
Chapter 014 — Eigenvectors: What a Transformation Leaves Alone → -
015
Turn, stretch, turn
Eigenvectors run out — a rotation has none, and a non-square matrix cannot have any. The singular value decomposition never does. Every matrix at all is a turn, a stretch along two perpendicular axes, and another turn.
The three steps · Keeping one axis · Always, even here · Strong vs weak directions
Chapter 015 — Singular Value Decomposition → -
016
The direction that loses least
A cloud of data has a direction it is mostly spread along. Finding it is not a new idea — it is the eigenvector of the covariance matrix, which is to say the direction that survives best when everything else is thrown away.
Step zero: centre it · Variance along a line · Project and rebuild · What compression loses
Chapter 016 — PCA: Compressing Data by Finding Important Directions →
More studios arrive with the chapters they belong to. The engine is shared, so a new one is a scene list rather than a new application.