CHAPTER 03 / 40
Analytic Geometry
8 lessons · Explore, experiment and understand.
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Inner products, norms and projections are how models measure similarity, build kernels, and decide which class a point belongs to.
- k-NN
- cosine similarity
- kernel methods
- Step 10~25m3D · 136★Most cinematic⚡Challenge
Norms
A norm on a vector space is a function that assigns each vector a non-negative length. Formally, it must satisfy three axioms: absolute homo…
Open lesson - Step 11~20m35⚡Challenge
Inner Products
An inner product is a function that eats two vectors and returns a scalar, satisfying three properties: symmetry (), linearity in each argum…
Open lesson - Step 12~20m25⚡Challenge
Lengths and Distances
Once we have a norm, we automatically get a distance between any two points: . The distance inherits all the good behavior of the norm — non…
Open lesson - Step 13~20m25⚡Challenge
Angles and Orthogonality
From an inner product we can extract the angle between two non-zero vectors via . Cauchy–Schwarz guarantees this ratio lies in , so is well-…
Open lesson - Step 14~20m25⚡Challenge
Orthonormal Basis
A set of vectors is orthonormal if each has unit length () and any two are orthogonal ( for ). Compactly, (the Kronecker delta). An orthonor…
Open lesson - Step 15~20m3D · 115⚡Challenge
Orthogonal Complement
Given a subspace of an inner-product space , its orthogonal complement is the set — every vector that is orthogonal to every vector in . Thi…
Open lesson - Step 16~20m3D · 225★Most cinematic⚡Challenge
Orthogonal Projections
An orthogonal projection onto a subspace is the linear map that sends each vector to its unique best approximation by an element of — the ve…
Open lesson - Step 17~20m3D · 215★Most cinematic⚡Challenge
Rotations
A rotation is a linear transformation that preserves lengths, angles, and orientation. In , rotation by angle counterclockwise is the matrix…
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