Fall 2014
- MATH 304-503: Linear Algebra
Time and venue: MWF 11:30 a.m.–12:20 p.m., BLOC 160
Office hours (BLOC 223b):
- Monday, 1:00–3:00 p.m.
- Wednesday, 1:00–2:00 p.m.
- by appointment
Help sessions (BLOC 117):
- Sunday-Monday, Wednesday-Thursday, 5:00–7:30 p.m.
Additional office hours (BLOC 223b):
- Friday, December 12, 11:00 a.m.–2:00 p.m.
- Monday, December 15, 11:00 a.m.–1:00 p.m.
- Tuesday, December 16, 11:00 a.m.–1:00 p.m.
Quiz 1: Wednesday, December 3 (solution)
Quiz 2: Friday, December 5 (solution)
Final exam: Wednesday, December 17, 10:30 a.m.-12:30 p.m., BLOC 160
Rules for the test: 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's book: Chapters 1-2
Lecture 1: Systems of linear equations.
Lecture 2: Gaussian elimination.
Lecture 3: Row echelon form. Gauss-Jordan reduction.
Lecture 4: System with a parameter. Applications of systems of linear equations.
Lecture 5: Matrix algebra.
Lecture 6: Diagonal matrices. Inverse matrix.
Lecture 7: Inverse matrix (continued).
Lecture 8: Elementary matrices. Transpose of a matrix. Determinants.
Lecture 9: Properties of determinants.
Lecture 10: 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's book: Chapters 3-4
Lecture 11: Vector spaces.
Lecture 12: Subspaces of vector spaces.
Lecture 13: Span. Spanning set.
Lecture 14: Linear independence.
Lecture 15: Wronskian. The Vandermonde determinant. Basis of a vector space.
Lecture 16: Basis and dimension.
Lecture 17: Basis and dimension (continued). Rank of a matrix.
Lecture 18: Nullity of a matrix. Basis and coordinates. Change of coordinates.
Lecture 19: Review for Test 1.
- Leon 1.1-1.5, 2.1-2.2, 3.1-3.4, 3.6
Lecture 20: Change of coordinates (continued). Linear transformations.
Lecture 21: Properties of linear transformations. Range and kernel. General linear equations.
Lecture 22: General linear equations (continued). Matrix transformations. Matrix of a linear transformation.
Lecture 23: Matrix of a linear transformation (continued). Similar matrices.
Part III: Advanced linear algebra
- Orthogonality
- Inner products and norms
- The Gram-Schmidt orthogonalization process
- Eigenvalues and eigenvectors
- Diagonalization
Leon's book: Sections 5.1-5.6, 6.1, 6.3
Lecture 24: Euclidean structure in Rn.
Lecture 25: Orthogonal complement. Orthogonal projection.
Lecture 26: Orthogonal projection (continued). Least squares problems.
Lecture 27: Norms and inner products.
Lecture 28: Inner product spaces. Orthogonal sets.
Lecture 29: Orthogonal bases. The Gram-Schmidt orthogonalization process.
Lecture 30: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
Lecture 31: Eigenvalues and eigenvectors (continued). Characteristic equation.
Lecture 32: Eigenvalues and eigenvectors of a linear operator.
Lecture 33: Basis of eigenvectors. Diagonalization.
Lecture 34: Review for Test 2.
- Leon 3.5, 4.1-4.3, 5.1-5.6, 6.1, 6.3
Part IV: Topics in applied linear algebra
- Matrix exponentials
- Rotations in space
- Orthogonal polynomials
- Markov chains
Leon's book: Sections 5.5, 5.7, 6.1-6.4
Lecture 35: Matrix polynomials. Matrix exponentials.
Lecture 36: Complex eigenvalues and eigenvectors. Symmetric and orthogonal matrices.
Lecture 37: Orthogonal matrices (continued). Rigid motions. Rotations in space.
Lecture 38: Orthogonal polynomials.
Lecture 39: Markov chains.
Lecture 40: Review for the final exam.
- Leon 1.1-1.5, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.7, 6.1-6.3