Fall 2009
MATH 311-505: Topics in Applied Mathematics
Time and venue: TR 11:10 a.m.-12:25 p.m., MILN 216
Office hours:
Wednesday, 10 a.m.-12 p.m.
Thursday, 10 a.m.-11 a.m.
by appointment
Office hours for the week of December 7:
Tuesday, 10 a.m.-11 a.m.
Wednesday, 10 a.m.-1 p.m.
Thursday, 10 a.m.-1 p.m.
Final exam: Friday, December 11, 3:00-5:00 p.m., MILN 216
Course outline:
Part I: Elementary linear algebra
Systems of linear equations
Matrices
Determinants
Leon's book: Chapters 1-2
Lecture 1: Systems of linear equations.
Leon 1.1
Lecture 2: Gaussian elimination. Row echelon form. Gauss-Jordan reduction.
Leon 1.1-1.2
Lecture 3: Some applications of systems of linear equations. Matrix algebra.
Leon 1.2-1.3
Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
Leon 1.3-1.4
Lecture 5: Inverse matrix (continued). Determinants.
Leon 1.4, 2.1
Lecture 6: Evaluation of determinants.
Leon 2.1-2.2
Part II: Abstract linear algebra
Vector spaces
Linear independence
Basis and dimension
Change of basis
Linear transformations
Leon's book: Chapters 3-4
Lecture 7: Vector spaces. Subspaces.
Leon 3.1-3.2
Lecture 8: Span. Spanning set.
Leon 3.2
Lecture 9: Linear independence. Basis of a vector space.
Leon 3.3-3.4
Lecture 10: Basis and dimension.
Leon 3.4
Lecture 11: Rank and nullity of a matrix. Basis and coordinates. Change of basis.
Leon 3.5-3.6
Lecture 12: Change of coordinates (continued). Review for Test 1.
Leon 3.5 (change of coordinates), 1.1-1.4, 2.1-2.2, 3.1-3.4, 3.6 (review for Test 1)
Lecture 13: Linear transformations. General linear equations. Matrix transformations.
Leon 4.1-4.2
Lecture 14a: Matrix of a linear transformation.
Leon 4.2-4.3
Part III: Advanced linear algebra
Eigenvalues and eigenvectors
Diagonalization
Orthogonality
Inner products and norms
The Gram-Schmidt orthogonalization process
Leon's book: Chapters 5-6 (selected sections)
Lecture 14b: Eigenvalues and eigenvectors.
Leon 6.1
Lecture 15: Eigenvalues and eigenvectors (continued).
Leon 6.1, 6.3
Lecture 16: Diagonalization. Euclidean structure in Rn.
Leon 6.2-6.3, 5.1
Lecture 17: Orthogonal complement. Orthogonal projection.
Leon 5.1-5.2
Lecture 18: Least squares problems. Norms and inner products.
Leon 5.3-5.4
Lecture 19: Orthogonal sets. The Gram-Schmidt orthogonalization process.
Leon 5.5-5.6
Lecture 20: The Gram-Schmidt process (continued). Review for Test 2.
Leon 3.5, 4.1-4.3, 5.1-5.6, 6.1, 6.3
Part IV: Applied linear algebra
Boundary value problems
Heat equation
Fourier series
Bessel functions
Spiegel's book: Chapters 1, 2, 6
Lecture 21: Boundary value problems. Separation of variables.
Spiegel, Ch. 1
Lecture 22: Fourier's solution of the heat equation. Fourier series.
Spiegel, Ch. 1-2
Lecture 23: Fourier series (continued).
Spiegel, Ch. 2
Lecture 24: Heat equation (continued). Bessel functions.
Spiegel, Ch. 2, 6
Lecture 25: Bessel functions (continued).
Spiegel, Ch. 6
Lecture 26: Review for the final exam.
Leon 1.1-1.4, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.6, 6.1, 6.3
Spiegel, Ch. 1, 2, 6