Linear Algebra

Preface to the Instructor
Preface to the Student
Acknowledgements
Notation

1.Matrices and Vectors. 1.1 Matrices and Linear Systems. 1.2 Row Reduction: Three Elementary Row Operations. 1.3 Vectors in Rn, Linear Combinations and Span. 1.4 Matrix Vector Product and the Equation Ax=b. 1.5 How to Check Your Work. 1.6 Exercises.

2.Subspaces in Rn, Basis and Dimension. 2.1 Subspaces in Rn. 2.2 Column Spaces, Row Spaces and Null Space of a Matrix. 2.3 Linear Independence. 2.4 Basis. 2.5 Coordinate Systems. 2.6 Exercises.

3.Matrix Algebra. 3.1 Matrix Addition and Multiplication. Transpose. Inverse. Elementary Matrices. Block Matrices. Lower and Upper Triangular Matrices and LU Factorization 3.7 Exercises

4.Determinants. Definition of the Determinant and Properties. Alternative Definition and Proofs of Properties. Cramer's rule. Determinants and Volumes. Exercises

5.Vector spaces. Definition of a Vector Space. Main Examples. Linear Independence, Span, and Basis. Coordinate Systems. Exercises.

6.Linear Transformations. Definition of a Linear Transformation. Range and Kernel of Linear Transformations. Matrix Representations of Linear Maps. Change of Basis. Exercises.

7.Eigenvectors and Eigenvalues. Eigenvectors and Eigenvalues. Similarity and Diagonalizability. Complex eigenvalues. Exercises.

8.Orthogonality. Dot Product and the Euclidean Norm. Orthogonality and Distance to Subspaces. Orthonormal Bases and Gram-Schmidt. Unitary Matrices and QR Factorization. Least Squares Solution and Curve Fitting. Real Symmetric and Hermitian Matrices.

Appendix.A1.1 Some Thoughts on Writing Proofs. A1.2 Mathematical Examples. A 1.3 Truth Tables. A1.4 Quantifiers and Negation of Statements. A1.5 Proof by Induction. A1.6 Some Final Thoughts. A.2 Complex numbers. A.3 The Field Axioms.