Linear algebra notes

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Title page 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 2 Polynomials over a field 2.1 Lagrange Interpolation Polynomials 2.2 Division of polynomials 2.2.1 Euclid's Division Theorem 2.2.2 Euclid's Division Algorithm 2.3 Irreducible Polynomials 2.4 Minimum Polynomial of a (Square) Matrix 2.5 Construction of a field of p^n elements 2.6 Characteristic and Minimum Polynomial of a Transformation 2.6.1 M_{nxn}(F[x])-Ring of Polynomial Matrices 2.6.2 M_{nxn}(F)[y]-Ring of Matrix Polynomials 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 m_T 3.3 Primary Decomposition Theorem 4 The Jordan Canonical Form 4.1 The Matthews' dot diagram 4.2 Two Jordan Canonical Form Examples 4.2.1 Example (a): 4.2.2 Example (b): 4.3 Uniqueness of the Jordan form 4.4 Non-derogatory matrices and transformations 4.5 Calculating A^m, where A∈M_{nxn}(C) 4.6 Calculating e^A, where A∈M_{nxn(}C) 4.7 Properties of the exponential of a complex matrix 4.8 Systems of differential equations 4.9 Markov matrices 4.10 The Real Jordan Form 4.10.1 Motivation 4.10.2 Determining the real Jordan form 4.10.3 A real algorithm for finding the real Jordan form 5 The Rational Canonical Form 5.1 Uniqueness of the Rational Canonical Form 5.2 Deductions from the Rational Canonical Form 5.3 Elementary divisors and invariant factors 5.3.1 Elementary Divisors 5.3.2 Invariant Factors 6 The Smith Canonical Form 6.1 Equivalence of Polynomial Matrices 6.1.1 Determinantal Divisors 6.2 Smith Canonical Form 6.2.1 Uniqueness of the Smith Canonical Form 6.3 Invariant factors of a polynomial matrix 7 Various Applications of Rational Canonical Forms 7.1 An Application to commuting transformations 7.2 Tensor products and the Byrnes-Gauger theorem 7.2.1 Properties of the tensor product of matrices 8 Further directions in linear algebra