Vector Spaces, Subspaces, Bases, Dimension

GRE Math Subject · Axiom Academy

LESSON Vector Spaces, Subspaces, Bases, Dimension The foundational structures of linear algebra with rigorous definitions and essential examples A vector space over a field (typically or ) is a set with two operations — addition and scalar multiplication — satisfying 10 axioms: Key insight: These axioms define the algebraic structure that makes linear combinations and linear independence meaningful. Recognize these fundamental examples: — all -tuples of real numbers (most common) — continuous real-valued functions on Counterexample: The set of vectors with first coordinate equal to 1 is NOT a vector space (no zero vector). A subset is a subspace if and only if: Intuition: You don't need to verify all 10 axioms for a subspace — just these three! Example: In , the -plane is a subspace. Vectors are linearly independent if the only solution to is . Otherwise they are linearly dependent . A basis for a vector space is a linearly independent set that spans . Example: The standard basis for is If a vector space has a basis with vectors, then: GRE Tip: Questions often ask you to verify subspaces, find bases, or compute dimensions. Always check closure properties explicitly.

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