#### Matrix Theory Math 314 Fall 2007 Section 001 Class Log

 1. Aug 27, Mon. course overview and logistics; Sec 2.1: systems of linear equations; solving via algebraic manipulations; Sec 2.2: matrix of coefficients, augmented matrix, elementary row operations, row echelon form, reduced row echelon form, reading off solutions 2. Aug 29, Wed. number of solutions; rank; Sec 1.1: vector review; Sec 2.3: vector equations; linear combinations; span of a set of vectors; linearly dependent set of vectors 3. Aug 31, Fri. more on linearly dependent sets of vectors; Sec 2.4: balancing chemical reactions; Quiz 1 Sept 3, Mon. Labor Day -- no class 4. Sept 5, Wed. electrical networks; Sec 3.1: matrices as mathematical objects; addition and scalar multiplication; transpose; matrix multiplication 5. Sept 7, Fri. Sec 3.2: block multiplication; properties of matrix addition, scalar multiplication, and multiplication; Quiz 2 6. Sept 10, Mon. Sec 3.3: invertible matrices, properties of invertible matrices, elementary matrices, propertices of elementary matrices 7. Sept 12, Wed. testing for invertibility; finding the inverse of an invertible matrix; Sec 3.3: definition of subspace; subspaces of R2; span of a set of vectors is a subspace 8. Sept 14, Fri. row space, column space, and null space of a matrix; testing if a vector is in these spaces; Quiz 3 9. Sept 17, Mon. definition of basis; finding bases for the row space, column space, and null space of a matrix 10. Sept 19, Wed. definition of dimension; rank of a matrix; nullity of a matrix; Fundamental Theorem of Invertible Matrices 11. Sept 21, Fri. coordinates with respect to a basis; Sec 6.3: change-of-basis matrix; Quiz 4 12. Sept 24, Mon. more on change-of-basis matrices; invertibility; computing using Gaussian elimination 13. Sept 26, Wed. Sec 3.6: linear transformations; linear transformations as matrix multiplication; rotation and projection; composition of functions 14. Sept 28, Fri. composition of functions; identity; inverse of linear transformations; Quiz 5 15. Oct 1, Mon. change-of-basis is an invertible linear transformation; Sec 3.7: applications: adjacency matrix of a graph; powers of the adjacency matrix; modeling the game of Monopoly; probability vectors; stationary distribution 16. Oct 3, Wed. Markov chains; mouse in maze example; Sec 4.1: eigenvalues and eigenvectors 17. Oct 5, Fri. Test 1 18. Oct 8, Mon. eigenvalues, eigenvectors, and eigenspaces; geometric interpretation; Sec 4.2: definition of determinant for 1x1, 2x2, 3x3 and nxn matrices 19. Oct 10, Wed. mnemenoic for evaluating 3x3 determinants; Laplace expansion theorem; properties of determinant, including relation to elementary row operations 20. Oct 12, Fri. returned Test 1; determinants of elementary matrices; using determinant to test for invertibility; Quiz 6 21. Oct 15, Mon. multiplicativity of determinants; Cramer's Rule; computing the inverse using the adjoint 22. Oct 17, Wed. applications of the determinant: Kirchoff Matrix Tree Theorem; defining area and volume; Jacobian for change of variables for multivariate integrals; Sec 4.3: using the determinant to find eigenvalues; characteristic equation; algebraic and geometric multiplicity 23. Oct 19, Fri. eigenvalues can be complex numbers; eigenvalues of triangular matrices; testing for invertibility using eigenvalues; eigenvalues of powers of a matrix; Quiz 7 Oct 22, Mon. Fall Break -- no class 24. Oct 24, Wed. Example of proof-type question; computing powers of matrices; linear independence of eigenvectors corresponding to different eigenvalues; eigenvectors of diagonal matrices 25. Oct 26, Fri. Sec 4.4: similarity of matrices; similar matrices have the same determinant, char poly, eigenvalues, and rank; diagonalizability; Quiz 8 26. Oct 29, Mon. matrix is diagonalizable if it has n lin indep eigenvectors; determining if a matrix is diagonalizable; diagonalization theorem; powers of a diagonalizable matrix 27. Oct 31, Wed. review for test; Sec 4.6: 1 is an eigenvalue of transition matrices for Markov chains; Google's PageRank algorithm 28. Nov 2, Fri. Test 2 29. Nov 5, Mon. Sec 1.2: dot product; norm; basic properties; Sec 5.1: orthogonal sets; orthogonal bases; finding coordinates; orthonormal bases 30. Nov 7, Wed. orthogonal matrices; properties; example of rotation matrices; preserves length and dot product; Sec 5.2: orthogonal complement; is a subspace; basic properties 31. Nov 9, Fri. returned Test 2; rowspace and nullspace are orthogonal complements; projection onto a subspace and perpendicular vector of a subspace; Quiz 9 32. Nov 12, Mon. distributed project; orthogonal decomposition theorem; dimensions of subspace and orthogonal complement; application to subspaces associated to a matrix 33. Nov 14, Wed. Sec 7.3: best approximation to a subspace; Sec 5.3: Gram-Schmidt process 34. Nov 16, Fri. Sec 5.4: orthogonal diagonalization; orthogonally diagonalizable matrices are symmetric; symmetric matrices have real eigenvalues; Quiz 10 35. Nov 19, Mon. finished Spectral Thm for real symmetric matrices; demonstration of Maple Nov 21-23 Thanksgiving Break -- no class 36. Nov 26, Mon. Sec 6.1: abstract vector spaces; examples; relation of addition and scalar multiplication 37. Nov 28, Wed. span of a set of vectors; Sec 6.2: linear independence; bases; coordinates with respect to a basis; identification with R^k; every basis of V has the same size 38. Nov 30, Fri. Sec 6.3: change of basis; Quiz 11 39. Dec 3, Mon. Sec 6.4: linear transformations of abstract vector spaces; examples; Sec 6.5: kernel and range of a linear transformation; evaluations 40. Dec 5, Wed. a linear transformation is determined by its action on a basis; examples; nullity(T)+rank(T)=dim V 41. Dec 7, Fri. Test 3 42. Dec 10, Mon. Sec 7.3: least squares approximation 43. Dec 12, Wed. orthogonal projection matrix in R^n; Sec 7.1: definition of inner product; Sec 7.5: inner product on space of continuous functions on a closed interval; projecting onto the subspace of polynomials 44. Dec 14, Fri. orthogonal basis for the subspace of polynomials of degree at most 2 on a closed interval; trigonometric polynomials and Fourier series; Quiz 12 Dec 20, Thurs, 10am-noon. Final Exam