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Applied Linear Algebra Carl de Boor draft 15jan03 i ii TABLE OF CONTENTS overview . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Sets, assignments, lists, and maps Sets . . . . . . . . . . . . . . . . . . . . . . . . . Assignments . . . . . . . . . . . . . . . . . . . . . Matrices . . . . . . . . . . . . . . . . . . . . . . . Lists of lists . . . . . . . . . . . . . . . . . . . . . Maps . . . . . . . . . . . . . . . . . . . . . . . . 1-1 and onto . . . . . . . . . . . . . . . . . . . . . Some examples . . . . . . . . . . . . . . . . . . . . Maps and their graphs . . . . . . . . . . . . . . . . . Invertibility . . . . . . . . . . . . . . . . . . . . . The inversion of maps . . . . . . . . . . . . . . . . . 2. Vector spaces and linear maps Vector spaces, especially spaces of functions . . . . . . . Linear maps . . . . . . . . . . . . . . . . . . . . . Linear maps from IFn . . . . . . . . . . . . . . . . . The linear equation A? = y, and ran A and null A . . . . . Inverses . . . . . . . . . . . . . . . . . . . . . . . 3. Elimination, or: The determination of null A and ran A Elimination and Backsubstitution . . . . . . . . . . . . The really reduced row echelon form and other reduced forms A complete description for null A obtained from a b-form . . The factorization A = A(:, bound)rrref(A) . . . . . . . . A ‘basis’ for ran A . . . . . . . . . . . . . . . . . . . Uniqueness of the rrref(A) . . . . . . . . . . . . . . . The rrref(A) and the solving of A? = y . . . . . . . . . The pigeonhole principle for square matrices . . . . . . . 4. The dimension of a vector space Bases . . . . . . . . . . . . . . . . . . . . . . . . Construction of a basis . . . . . . . . . . . . . . . . . Dimension . . . . . . . . . . . . . . . . . . . . . . Some uses of the dimension concept . . . . . . . . . . . The dimension of IFT . . . . . . . . . . . . . . . . .