Understand the abstract structures that unify linear algebra.
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Every linear transformation from ℝⁿ to ℝᵐ can be represented as multiplication by an m×n matrix. Geometric examples:
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A vector space V over ℝ is a set with addition and scalar multiplication satisfying 8 axioms (closure, associativity, commutativity, identity, inverse, compatibility, distributivity). Examples beyond ℝⁿ:
A subspace is a subset closed under addition and scalar multiplication. Important subspaces of a matrix A: column space (Col A), row space, null space (Nul A).
Change of basis transforms coordinates between different bases — essential when working with eigenvectors as a basis (diagonalization).