Fall 2015
- MATH 311-503: Topics in Applied Mathematics I
Time and venue: MWF 9:10–10:00 a.m., BLOC 163
Office hours (BLOC 223b):
- Monday, 12:00–1:00 p.m.
- Wednesday, 12:00–2:00 p.m.
- by appointment
Help sessions (BLOC 117):
- Sunday-Thursday, 8:00–10:00 p.m.
Additional office hours (BLOC 223b):
- Friday, December 11, 11:00 a.m.–1:00 p.m.
- Monday, December 14, 11:00 a.m.–1:00 p.m.
- Tuesday, December 15, 11:00 a.m.–1:00 p.m.
Final exam: Monday, December 14, 8:00-10:00 a.m., BLOC 163
Rules for the exam: no books, no lecture notes, no computing devices. Bring paper and a stapler.
Course outline:
Part I: Elementary linear algebra
- Systems of linear equations
- Gaussian elimination, Gauss-Jordan reduction
- Matrices, matrix algebra
- Determinants
Leon/Colley: Chapters 1-2
Lecture 1: Systems of linear equations.
Lecture 2: Gaussian elimination.
Lecture 3: Row echelon form. Gauss-Jordan reduction.
Lecture 4: Applications of systems of linear equations. Matrix algebra.
Lecture 5: Matrix multiplication. Diagonal matrices.
Lecture 6: Inverse matrix.
Lecture 7: Inverse matrix (continued). Transpose of a matrix.
Lecture 8: Properties of determinants.
Lecture 9: Evaluation of determinants. Cramer's rule.
Part II: Abstract linear algebra
- Vector spaces
- Linear independence
- Basis and dimension
- Coordinates, change of basis
- Linear transformations
Leon/Colley: Chapters 3-4
Lecture 10: Vector spaces.
Lecture 11: Subspaces of vector spaces.
Lecture 12: Span. Spanning set.
Lecture 13: Linear independence.
Lecture 14: Review for Test 1.
- Leon/Colley 1.1-1.5, 2.1-2.2, 3.1-3.3
Lecture 15: Basis and dimension.
Lecture 16: Basis and dimension (continued). Rank of a matrix.
Lecture 17: Nullity of a matrix. Basis and coordinates. Change of basis.
Lecture 18: Change of basis (continued). Linear transformations.
Lecture 19: Range and kernel. General linear equations. Matrix transformations.
Lecture 20: Matrix of a linear transformation. Similar matrices.
Part III: Advanced linear algebra
- Orthogonality
- Inner products and norms
- The Gram-Schmidt orthogonalization process
- Eigenvalues and eigenvectors
- Diagonalization
Leon/Colley: Chapters 5-7
Lecture 21: Eigenvalues and eigenvectors. Characteristic equation.
Lecture 22: Eigenvalues and eigenvectors of a linear operator.
Lecture 23: Basis of eigenvectors. Diagonalization.
Lecture 24: Orthogonal complement. Orthogonal projection.
Lecture 25: Orthogonal projection (continued). Least squares problems.
- Leon/Colley 5.2-5.3, 7.3, 7.5
Lecture 26: Least squares problems (continued). Orthogonal bases. The Gram-Schmidt orthogonalization process.
Lecture 27: Review for Test 2.
- Leon/Colley 3.4-3.6, 4.1-4.3, 5.1-5.3, 5.5-5.6, 6.1, 6.3
Lecture 28: Norms and inner products.
Lecture 29: Orthogonality in inner product spaces.
Part IV: Vector analysis
- Main notions of vector analysis
- Review of multiple integrals
- Line and surface integrals
- Green's, Gauss' and Stokes' theorems
Leon/Colley: Chapters 8-11
Lecture 30: Review of differential calculus.
Lecture 31: Differentiation in vector spaces. Gradient, divergence, and curl.
- Leon/Colley 7.4, 8.1, 8.3-8.4
Lecture 32: More on the differential. Review of integral calculus.
Lecture 33: Multiple integrals. Line integrals.
- Leon/Colley 8.2, 9.1-9.5, 10.1
Lecture 34: Green's theorem. Conservative vector fields.
Lecture 35: Area of a surface. Surface integrals.
Lecture 36: Gauss' theorem. Stokes' theorem.
Lecture 37: Review for Test 3.
- Leon/Colley 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3
Lecture 38: Review for the final exam.
- Leon/Colley 1.1-1.5, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.6, 6.1-6.3, 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3
Lecture 39: Integration of differential forms. Review for the final exam (continued).
- Leon/Colley 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3