Inner product
Element-wise products summed; relates to the angle between vectors.
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02
Foundations
Mathematical Foundations
The linear algebra and probability used by every model in the course.
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What you’ll take away
- Vectors hold examples, matrices hold datasets and layer weights; matrix products power predictions.
- Dot products measure alignment; L1 and L2 distances measure difference.
- Expectation and variance summarise distributions; covariance links features.
- Bayes' theorem turns priors and likelihoods into posteriors — beware small base rates.
Major concepts (syllabus)
Scalars, vectors, matrices, tensors; vector operations; inner products; matrix multiplication; norms and distances; linear transformations; probability; random variables; expectation; variance; covariance; conditional probability; Bayes' theorem
Lessons
- Scalars, vectors, matrices, tensorsObjects of 0, 1, 2 and more dimensions, and how data is stored in them.8 min
- Vector operationsAdding, scaling and combining vectors.7 min
- Inner productsThe dot product and what it says about angle and similarity.8 min
- Matrix multiplicationComposing linear maps; inner dimensions must match.9 min
- Manhattan distance (L1) SlidesSum of absolute coordinate differences — the 'taxicab' distance on a grid.9 min
- Euclidean distance (L2) SlidesThe straight-line distance between two points, ignoring paths and turns.8 min
- Linear transformations and the Jacobian SlidesMatrices as maps, and the Jacobian as the derivative of a vector-valued function.10 min
- Probability basicsSample spaces, events and the rules of probability.8 min
- Random variablesDiscrete and continuous variables and their distributions.8 min
- Expectation and varianceThe average value and the spread around it.9 min
- CovarianceHow two variables move together.7 min
- Conditional probabilityProbability of an event given that another has happened.8 min
- Bayes' theoremUpdating a prior belief with evidence to get a posterior.10 min
Key formulas
Manhattan distance
Slides
Sum of absolute differences (L1).
Euclidean distance
Slides
Straight-line distance (L2).
Lp norm
p = 1 Manhattan, p = 2 Euclidean, p → ∞ max-norm.
Jacobian matrix
Slides
Derivative of a vector-valued function; how it transforms local space.
Expectation
Probability-weighted average.
Variance
Expected squared distance from the mean.
Covariance
Positive when X and Y tend to rise together.
Bayes' theorem
Posterior = likelihood × prior / evidence.
Key terms
- Bayes' theorem
- P(H|E) = P(E|H)·P(H) / P(E): updating a prior belief with evidence.
- Covariance
- How two variables vary together; positive when they rise together.
- Euclidean distance
- Straight-line (L2) distance: square root of the sum of squared coordinate differences.
- Inner product
- Sum of element-wise products of two vectors; measures alignment.
- Jacobian
- Matrix of all first partial derivatives of a vector-valued function.
- Manhattan distance
- L1 distance: sum of absolute coordinate differences.
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