1.1 Vectors, spaces and geometric intuition
You can read a vector equation and see the picture behind it.
Before:00. Orientation & SetupUnlocks:03. Data Handling & Analysis04. Classical AI — Agents, Search & Knowledge Representation
Vectors are how every dataset, embedding and model parameter is represented, and this topic builds the geometric picture — points, arrows, projections — until it is second nature. The dot product as similarity is the single most reused idea in the subject, returning in attention and vector search. It opens the mathematics module because everything after speaks this language. The confusion to clear is treating vectors as mere lists of numbers; the geometry is what makes later concepts obvious instead of arbitrary.
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Vectors as points, arrows and lists of features
A vector can be pictured as a point in space, as an arrow with direction and length, or as a plain list of numbers, and machine learning uses all three readings interchangeably. Being able to switch between them is what makes later notation readable.
Dot product: projection, similarity, cosine
The dot product measures how much one vector points along another, which is why it appears as similarity everywhere from recommendation to attention. The cosine version removes the effect of length and keeps only direction.
Linear independence, span, basis, dimension
Whether a set of vectors adds genuinely new directions, how much space they cover between them, and the smallest set that does the job. These four words describe the same idea from different angles.
Linear Algebra (18.06SC) · CourseSubspaces and why data lives on a lower-dimensional one
Real data almost never fills the space it is described in; it concentrates near a smaller surface inside it. That observation is the reason dimensionality reduction works at all.
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