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The Rank Theorem
Definition: Let A be the coefficient matrix of a system of linear equations with n variables. If the system is consistent, then: number of free variables = n – rank(A). In other words, if A is an m X n matrix, then rank(A) + nullity(A) = n.
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)