Learning lesson 02 · Turn data into geometry
01 · Smallest useful mechanism
A feature vector is an ordered list such as [size, brightness]. A weight vector is another ordered list in the same coordinate system. Multiplying matching coordinates and summing them produces a dot product. Add a bias and the scalar score becomes a linear decision. Rotating or scaling the weight vector changes which evidence the model treats as important.
The dot product combines input features with a weight direction; its sign and magnitude determine which side of a linear boundary the example occupies.
02 · Experiment
Calculate by hand · two-dimensional vectors
The small coordinates make every dot product and decision inspectable. Real neural vectors use many more dimensions but obey the same shape rules.
Weight direction
The green arrow is w. The zero-score boundary stays perpendicular to it.
| Example | Vector x | w₁x₁ | w₂x₂ | + bias | score | class |
|---|---|---|---|---|---|---|
| A | [0.8, 0.2] | 0.64 | -0.08 | 0.00 | 0.56 | +1 |
| B | [0.3, 0.9] | 0.24 | -0.36 | 0.00 | -0.12 | −1 |
| C | [-0.7, 0.5] | -0.56 | -0.20 | 0.00 | -0.76 | −1 |
| D | [-0.4, -0.8] | -0.32 | 0.32 | 0.00 | 0.00 | −1 |
03 · Reason from evidence
Your brief: Audit the representation contract before blaming the model. Explain how unchanged parameters can produce different scores.
Evidence your answer should use
There is no reveal button here. A strong response names the claim, the evidence that would test it, and what the evidence cannot establish.
Your learning artifact
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04 · Check your understanding
Next: A score becomes useful for learning only after we define how costly a prediction should be.
