Vector Spaces Summary

Linear Algebra (Matrices) · Axiom Academy

Unit 4 in one place: vectors and their operations, span and independence, subspaces, and the basis that pins down a space's dimension. A vector in is built up from others through linear combinations — scaling and adding. Everything else in this unit grows from that one move. Span is every linear combination you can reach; linear independence says no vector in the set is redundant. A set that is both spanning and independent is a basis . The number of vectors in a basis is the dimension — and it is the same for every basis of the space, so dimension is a true property of the space, not of the basis you picked. Subspaces (a set closed under addition and scaling that contains ) appear naturally as a matrix's column space and null space . An n -vector is an ordered list of n real numbers — a point, a displacement, or a direction in n -dimensional space. We work in columns by default. Special role: the zero vector is the additive identity and lives in every subspace. Watch out for: a row vs. column is a notation choice; keep it consistent within a problem. Core Concept Operations & the Dot Product Addition and scalar multiplication act component-wise — the two operations that define a vector space. The dot product collapses two vectors to a single number measuring alignment. Orthogonality: means the vectors meet at right angles, e.g. . Watch out for: the dot product returns a scalar, not a vector. Core Concept Linear Combinations & Span

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