Spring 2010
MATH 304-503: Linear Algebra
Time and venue: TR 12:45-2:00 p.m., MILN 317
Office hours:
Tuesday, 11:00 a.m.-12:30 p.m.
Thursday, 11:00 a.m.-12:30 p.m.
by appointment
Help sessions:
Mondays - Thursdays, 7:00-9:30 p.m., ENPH 215
Office hours before the final exam:
Tuesday, May 4, 11:00 a.m.–1:00 p.m.
Thursday, May 6, 11:00 a.m.–1:00 p.m.
Tuesday, May 11, 2:00–4:00 p.m.
Final exam: Wednesday, May 12, 8:00–10:00 a.m., MILN 317
Rules for the exam: no books, no lecture notes, no calculators.
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.
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
Lecture 5: Inverse matrix (continued).
Leon 1.3-1.4
Lecture 6: Transpose of a matrix. Determinants.
Leon 1.3, 2.1-2.2
Lecture 7: Evaluation of determinants. Cramer's rule.
Leon 2.1-2.3
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 8: Vector spaces. Subspaces.
Leon 3.1-3.2
Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
Leon 3.2
Lecture 10: Linear independence. Basis of a vector space.
Leon 3.3-3.4
Lecture 11: Basis and dimension.
Leon 3.4
Lecture 12: Rank and nullity of a matrix.
Leon 3.2, 3.6
Lecture 13: Review for Test 1.
Leon 1.1-1.4, 2.1-2.2, 3.1-3.4, 3.6
Lecture 14: Basis and coordinates. Change of basis. Linear transformations.
Leon 3.5, 4.1
Lecture 15: Kernel and range. General linear equation. Matrix transformations.
Leon 4.1-4.2
Lecture 16a: Matrix of a linear transformation. Similar matrices.
Leon 4.2-4.3
Part III: Advanced linear algebra
Orthogonality
Inner products and norms
The Gram-Schmidt orthogonalization process
Eigenvalues and eigenvectors
Diagonalization
Leon's book: Chapters 5-6 (selected sections)
Lecture 16b: Euclidean structure in Rn.
Leon 5.1
Lecture 17: Euclidean structure in Rn (continued). Orthogonal complement. Orthogonal projection.
Leon 5.1-5.2
Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vetor spaces.
Leon 5.2-5.4
Lecture 19: Inner product spaces. Orthogonal sets. The Gram-Schmidt process.
Leon 5.4-5.6
Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
Leon 5.5-5.6, 6.1
Lecture 21: Eigenvalues and eigenvectors (continued). Characteristic polynomial.
Leon 6.1, 6.3
Lecture 22: Diagonalization. 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
Fourier series
Markov chains
Leon's book: Chapters 5-6 (selected sections)
Lecture 23: Matrix exponentials.
Leon 6.2-6.3
Lecture 24: Complexification. Orthogonal matrices. Rotations in space.
Leon 5.5, 6.1, 6.3-6.4
Lecture 25: Orthogonal polynomials.
Leon 5.7
Lecture 26: Review for the final exam.
Leon 1.1-1.4, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.7, 6.1-6.3