Maths 2.2Lesson outlineLesson 2 of 5 in Module 2
Vector Arithmetic and Norms
Add, scale and measure vectors, and know that "length" has more than one definition.
Video in production.
Learn
In this lesson11 parts
- 01Which car is most like this one?
- 02Adding, scaling and subtracting vectors
- 03The mean of vectors
- 04Length three ways: the L2, L1 and L∞ norms
- 05Distance as the norm of a difference
- 06Unit balls for the three norms
- 07Checkpoint: arithmetic, norms and distance
- 08The nearest car, raw and standardised
- 09Norms as penalties on the weights
- 10Vector arithmetic and norms in NumPy
- 11Recap, challenge, and next: the dot product
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- vector addition
- add matching components; head to tail |
(3, 1) + (1, 2) = (4, 3) - scalar multiplication
- every component times one number |
2 · (3, 1) = (6, 2) - mean vector (centroid)
- the component-by-component mean of a set of points |
(4, 3) - norm
- a length of a vector |
- L2 (Euclidean) norm
√(Σ vᵢ²), the straight-line length |5for(3, −4)- L1 (Manhattan) norm
Σ \ | vᵢ\ ||7- L∞ (max) norm
max \ | vᵢ\ ||4- distance
- the norm of the difference,
‖a − b‖|2.2361from(3, 1)to(1, 2) - unit ball
- every point with norm at most one | circle, diamond, square
- standardising
- subtract the mean, divide by the standard deviation | the lot's age and mileage
- z score
- a value in standard deviations from the mean | car 25's age,
−0.19 - penalty
- a norm of the weights added to a model's error | ridge:
‖w‖₂²; lasso:‖w‖₁
Quiz
Your first pick on each question is the one that counts, and a right one earns a coin. Getting one wrong here is how the lesson sticks.