Nicholas A. Loehr received his Ph.D. in mathematics from the University of California at San Diego in 2003, studying algebraic combinatorics under the guidance of Professor Jeffrey Remmel. After spending two years at the University of Pennsylvania as an NSF postdoc, Dr. Loehr taught mathematics at the College of William and Mary, the United States Naval Academy, and Virginia Tech. Dr. Loehr has authored over sixty refereed journal articles and three textbooks on combinatorics, advanced linear algebra, and mathematical proofs. He teaches classes in these subjects and many others, including cryptography, vector calculus, modern algebra, real analysis, complex analysis, and number theory.

• 1. Overview of Algebraic Systems. 2. Permutations. 3. Polynomials. 4. Basic Matrix Operations. 5. Determinants via Calculations. 6. Comparing Concrete Linear Algebra to Abstract Linear Algebra. 7. Hermitian, Positive Definite, Unitary, and Normal Matrices. 8. Jordan Canonical Forms. 9. Matrix Factorizations. 10. Iterative Algorithms in Numerical Linear Algebra. 11. Affine Geometry and Convexity. 12. Ruler and Compass Constructions. 13. Dual Vector Spaces. 14. Bilinear Forms. 15. Metric Spaces and Hilbert Spaces. 16. Finitely Generated Commutative Groups. 17. Introduction to Modules. 18. Principal Ideal Domains, Modules over PIDs, and Canonical Forms. 19. Introduction to Universal Mapping Properties. 20. Universal Mapping Problems in Multilinear Algebra

Nicholas A. Loehr received his Ph.D. in mathematics from the University of California at San Diego in 2003, studying algebraic combinatorics under the guidance of Professor Jeffrey Remmel. After spending two years at the University of Pennsylvania as an NSF postdoc, Dr. Loehr taught mathematics at the College of William and Mary, the United States Naval Academy, and Virginia Tech. Dr. Loehr has authored over sixty refereed journal articles and three textbooks on combinatorics, advanced linear algebra, and mathematical proofs. He teaches classes in these subjects and many others, including cryptography, vector calculus, modern algebra, real analysis, complex analysis, and number theory.

• 1. Overview of Algebraic Systems. 2. Permutations. 3. Polynomials. 4. Basic Matrix Operations. 5. Determinants via Calculations. 6. Comparing Concrete Linear Algebra to Abstract Linear Algebra. 7. Hermitian, Positive Definite, Unitary, and Normal Matrices. 8. Jordan Canonical Forms. 9. Matrix Factorizations. 10. Iterative Algorithms in Numerical Linear Algebra. 11. Affine Geometry and Convexity. 12. Ruler and Compass Constructions. 13. Dual Vector Spaces. 14. Bilinear Forms. 15. Metric Spaces and Hilbert Spaces. 16. Finitely Generated Commutative Groups. 17. Introduction to Modules. 18. Principal Ideal Domains, Modules over PIDs, and Canonical Forms. 19. Introduction to Universal Mapping Properties. 20. Universal Mapping Problems in Multilinear Algebra

Nicholas A. Loehr received his Ph.D. in mathematics from the University of California at San Diego in 2003, studying algebraic combinatorics under the guidance of Professor Jeffrey Remmel. After spending two years at the University of Pennsylvania as an NSF postdoc, Dr. Loehr taught mathematics at the College of William and Mary, the United States Naval Academy, and Virginia Tech. Dr. Loehr has authored over sixty refereed journal articles and three textbooks on combinatorics, advanced linear algebra, and mathematical proofs. He teaches classes in these subjects and many others, including cryptography, vector calculus, modern algebra, real analysis, complex analysis, and number theory.

• 1. Overview of Algebraic Systems. 2. Permutations. 3. Polynomials. 4. Basic Matrix Operations. 5. Determinants via Calculations. 6. Comparing Concrete Linear Algebra to Abstract Linear Algebra. 7. Hermitian, Positive Definite, Unitary, and Normal Matrices. 8. Jordan Canonical Forms. 9. Matrix Factorizations. 10. Iterative Algorithms in Numerical Linear Algebra. 11. Affine Geometry and Convexity. 12. Ruler and Compass Constructions. 13. Dual Vector Spaces. 14. Bilinear Forms. 15. Metric Spaces and Hilbert Spaces. 16. Finitely Generated Commutative Groups. 17. Introduction to Modules. 18. Principal Ideal Domains, Modules over PIDs, and Canonical Forms. 19. Introduction to Universal Mapping Properties. 20. Universal Mapping Problems in Multilinear Algebra