Equations.
We know the scores. Now find the recipe.
Predict first
x² = 4 has exactly two answers. Can a system of linear equations have exactly two?
Linear
Nudge by one. Rise by the same, every time.
That is all linear means: the output climbs by the same step wherever you start, whatever the other unknown is doing.
Every recipe at once
One patient. A whole line of answers.
Patient A scored 8. Every point in this plane is a recipe. Ask them all at once, and watch which ones agree.
Drag the recipe
Walk the line
Explain one patient perfectly. Then meet the other.
Walk along A’s line and A stays explained at every step. Watch how wrong the second patient is, until it isn’t.
Planes
Every patient takes away a degree of freedom.
With three weights, each patient’s equation is a flat plane. Two planes meet in a line. Three meet in a point.
Drag to turn
Two worlds
One equation. Two pictures.
On the left, recipes are points and patients are lines. On the right, scores are points and features are arrows. A is the bridge.
Too many dials
Too few, nothing fits. Too many, anything does.
Ten patients. Add features one at a time, and watch the fit go from impossible, to perfect, to meaningless.
Real scoreBest recipe’s prediction
The prediction shown is the best recipe: the one that misses least, found by least squares (Lecture 12). Past ten dials, the shortest of the perfect ones.
Prove it
Four questions. The reasons matter more than the score.
Question 1 of 4
500 patients, 12 features, A x = b. How many numbers are in x, and how many in b?
Question 2 of 4
A model predicts risk as w₀ + w₁ · age + w₂ · age². Is it linear?
Question 3 of 4
Patient A’s score changes from 8 to 12. What happens to A’s line in the row picture?
Question 4 of 4
Two different recipes u and v both satisfy A x = b. What about the recipe exactly halfway, ½u + ½v?