LINEAR ALGEBRA NOTES MP274 1991 K. R. MATTHEWS LaTeXed by Chris Fama DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND 1991 Comments to the author at
[email protected] Contents 1 Linear Transformations 1.1 Rank + Nullity Theorems (for Linear Maps) 1.2 Matrix of a Linear Transformation . . . . . . 1.3 Isomorphisms . . . . . . . . . . . . . . . . . . 1.4 Change of Basis Theorem for TA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 6 12 18 2 Polynomials over a field 20 2.1 Lagrange Interpolation Polynomials . . . . . . . . . . . . . . 21 2.2 Division of polynomials . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Euclid’s Division Theorem . . . . . . . . . . . . . . . . 24 2.2.2 Euclid’s Division Algorithm . . . . . . . . . . . . . . . 25 2.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . 26 2.4 Minimum Polynomial of a (Square) Matrix . . . . . . . . . . 32 2.5 Construction of a field of pn elements . . . . . . . . . . . . . . 38 2.6 Characteristic and Minimum Polynomial of a Transformation 41 2.6.1 Mn×n (F [x])—Ring of Polynomial Matrices . . . . 42 2.6.2 Mn×n (F )[y]—Ring of Matrix Polynomials . . . . 43 3 Invariant subspaces 3.1 T –cyclic subspaces . . . . . . . . . . . . . . . . . . . 3.1.1 A nice proof of the Cayley-Hamilton theorem 3.2 An Algorithm for Finding mT . . . . . . . . . . . . . 3.3 Primary Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 54 57 58 61 4 The Jordan Canonical Form 4.1 The Matthews’ dot diagram . . . . . . . . . . . . . 4.2 Two Jordan Ca