Linear Algebra Summary

GRE Math Subject · Axiom Academy

SUMMARY Linear Algebra Summary Unit 5 Review: Key concepts, connections, and strategic insights Linear algebra is the study of vector spaces, linear transformations, and their matrix representations. It underpins nearly 30% of the GRE Math Subject Test. This unit covered the foundational theory and computational techniques essential for the exam. A vector space over field (usually ) is a set with addition and scalar multiplication satisfying 10 axioms. A subspace is a subset closed under these operations and containing the zero vector. Why it matters: Subspaces are the "natural domains" for linear transformations. You must quickly identify and verify subspaces. A basis is a linearly independent spanning set. Every vector space has a unique dimension (the size of any basis). Dimension determines how many "degrees of freedom" the space has. A linear transformation preserves linear combinations. Every linear transformation between finite-dimensional spaces has a matrix representation (which depends on the choice of bases). The rank-nullity theorem connects dimensions of the four fundamental subspaces: column space, null space, row space, left null space. Strategic insight: Row operations preserve rank, nullity, and row space—but not the column space! Use the original matrix to find column space basis. The determinant measures how a matrix scales volumes. A matrix is invertible iff . Swap rows → multiply det by -1 Add multiple of row → det unchanged

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