Vector spaces

A vector space is the setting in which linear combinations make sense. This chapter studies which subsets inherit that structure, which vectors a chosen set produces and when the coefficients are unique. It assumes row reduction of linear systems and works over $\mathbb R$ unless stated otherwise; every span, independence and dimension claim depends on the chosen scalar field.

Lessons

1. Vector spaces: the axioms, their first consequences and the coordinate, polynomial and matrix examples.
2. Subspaces: subsets that inherit the operations, tested by closure.
3. Spans: every linear combination of chosen vectors, with membership decided by a linear system.
4. Linear independence: when no vector is redundant, so coefficients are unique.
5. Basis and dimension: independent spanning sets and the number of vectors they contain.
6. Coordinates and bases: building bases from data and recording coefficients as coordinate vectors.
7. Data fitting: least-squares lines, and exact polynomial interpolation as a coordinate problem.
8. Seen questions: past exam questions across coordinate, polynomial, function and matrix spaces, with later-topic previews marked.

Where it leads

Linear transformations are the maps that preserve linear combinations. Once bases are chosen, coordinates turn such a map into a matrix.

Vocabulary checkpoints

• A span is a subspace; a spanning set is a list of vectors that generates it.
• Independence makes coefficients unique; a basis is also required to span.
• Coordinates depend on the chosen ordered basis; the vector itself does not.
• Interpolation fits the data exactly, while least squares only approximates it.