Linear Algebra

Lina Oliveira earned her DPhil in Mathematics from the University of Oxford, UK and is a faculty member of the Department of Mathematics of Instituto Superior T□ecnico, University of Lisbon, Portugal. She has taught several graduate and undergraduate courses at Instituto Superior T□ecnico where she regularly teaches Linear Algebra since 2001. Her research interests are in the areas of Functional Analysis, Operator Algebras and Operator Theory. She is currently the head of the Linear Algebra course at the portuguese Air Force Academy.

1. Matrices. 1.1. Real and Complex Matrices. 1.2. Matrix Calculus. 1.3. Matrix Inverses. 1.4. Elementary Matrices. 1.5. Exercises. 1.6. At a Glance. 2. Determinant. 2.1. Axiomatic Definition. 2.2. Leibniz's Formula. 2.3. Laplace's Formula. 2.4. Exercises. 2.5. At a Glance. 3. Vector Spaces. 3.1. Vector Spaces. 3.2. Linear Independence. 3.3. Bases and Dimension. 3.4. Null Space, Row Space and Column Space. 3.5. Sum and intersection of Subspaces. 3.6. Change of Basis. 3.7. Exercises. 3.8. At a Glance. 4. Eigenvalues and Eigenvectors. 4.1. Spectrum of a Matrix. 4.2. Spectral Properties. 4.3. Similarity and Diagonalisation. 4.4. Jordan Canonical Form. 4.5. Exercises. 4.6. At a Glance. 5. Linear Transformations. 5.1. Linear Transformations. 5.2. Matrix Representations. 5.3. Null Space and Image. 5.4. Isomorphisms and Rank-Nullity Theorem. 5.5. Composition and Invertibility. 5.6. Change of Basis. 5.7. Spectrum and Diagonalisation. 5.8. Exercises. 5.9. At a Glance. 6. Inner Product Spaces. 6.1. Real Inner Product Spaces. 6.2. Complex Inner Product Spaces. 6.3. Orthogonal Sets. 6.4. Orthogonal and Unitary Diagonalisation. 6.5. Singular Value decomposition. 6.6. Affine Subspaces of Rn. 6.7. Exercises. 6.8. At a Glance. 7 Special Matrices by Example. 7.1. Least Squares Solutions. 7.2. Markov Chains. 7.3. Population Dynamics. 7.4. Graphs. 7.5. Differential Equations. 7.6. Exercises. 7.7. At a Glance. 8. Appendix. 8.1. Uniqueness of Reduced Row Echelon Form. 8.2. Uniqueness of determinant. 8.3. Direct sum of Subspaces. 9. Solutions.