Rank-Nullity Theorem and Fundamental Subspaces

GRE Math Subject · Axiom Academy

LESSON Rank-Nullity Theorem and Fundamental Subspaces The four fundamental subspaces associated with every matrix The Four Fundamental Subspaces For an matrix , there are four fundamental subspaces: Key fact: Row space and null space are orthogonal complements in . Same for column space and left null space in . Example: For a matrix with rank 3: Finding Rank and Bases for Subspaces Reduce the matrix to row echelon form (REF). Then: Rank: Number of nonzero rows in REF Column space basis: Original columns corresponding to pivot columns Row space basis: Nonzero rows of REF (they span the same space) Null space basis: Solve with free variables Important: Row operations preserve the row space and null space, but change the column space. Use original matrix for column space basis! In , we have a fundamental decomposition: This means every vector can be uniquely written as: Find the rank of a matrix (row reduce) Determine if a vector is in the column space Find a basis for the null space Verify dimensions using rank-nullity Use orthogonal complements to solve problems Determine linear independence using rank Quick check: For an invertible matrix, (full rank) and .

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