Spring 2012
MATH 423-200/500: Linear Algebra II
Time and venue: MWF 12:40-1:30 p.m., BLOC 160
Office hours (MILN 004):
MWF 11:00 a.m.-12:00 p.m.
by appointment
Additional office hours (MILN 004):
Wednesday, May 2, 11:00 a.m.-1:00 p.m.
Thursday, May 3, 11:00 a.m.-1:00 p.m.
Monday, May 7, 12:00-3:00 p.m.
Quiz 1: Friday, April 13 (solution)
Quiz 2: Friday, April 20 (solutions)
Quiz 3: Friday, April 27 (solution)
Course outline:
- Vector spaces (2 weeks)
- Linear transformations (2 weeks)
- Systems of linear equations, matrix algebra, determinants (2 weeks)
- Eigenvalues and eigenvectors, diagonalization (2.5 weeks)
- Inner product spaces, special classes of operators (3.5 weeks)
- Jordan canonical form (1.5 weeks)
Lecture notes:
Lecture 1: Classical vectors. Vector space.
Friedberg/Insel/Spence 1.1-1.2
Lecture 2: Vector spaces: examples and basic properties.
Friedberg/Insel/Spence 1.2
Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field.
Friedberg/Insel/Spence 1.3, appendix C, appendix D
Lecture 4: Span. Spanning set. Linear independence.
Friedberg/Insel/Spence 1.3-1.5
Lecture 5: Linear independence (continued).
Friedberg/Insel/Spence 1.5
Lecture 6: Basis and dimension.
Friedberg/Insel/Spence 1.6-1.7
Lecture 7: Linear transformations. Range and null-space.
Friedberg/Insel/Spence 2.1
Lecture 8: Subspaces and linear transformations. Basis and coordinates. Matrix of a linear transformation.
Friedberg/Insel/Spence 2.1-2.2
Lecture 9: Matrix of a linear transformation (continued). Matrix multiplication.
Friedberg/Insel/Spence 2.2-2.3
Lecture 10: Inverse matrix. Change of coordinates.
Friedberg/Insel/Spence 2.3-2.5
Lecture 11: Change of coordinates (continued). Isomorphism of vector spaces.
Friedberg/Insel/Spence 2.4-2.5
Lecture 12: Review for Test 1.
Friedberg/Insel/Spence 1.1-1.7, 2.1-2.5, appendices C and D
Lecture 13: Advanced constructions of vector spaces.
Friedberg/Insel/Spence 1.3(exercises)
Lecture 14: General linear equations. Elementary matrices.
Friedberg/Insel/Spence 3.1
Lecture 15: Inverse matrix (continued). Transpose of a matrix.
Friedberg/Insel/Spence 3.2
Lecture 16: Rank of a matrix. Systems of linear equations. Reduced row echelon form.
Friedberg/Insel/Spence 3.2-3.4
Lecture 17: Reduced row echelon form (continued). Determinant of a matrix.
Friedberg/Insel/Spence 3.3-3.4, 4.1
Lecture 18: Determinants (continued).
Friedberg/Insel/Spence 4.1-4.4
Lecture 19: More on determinants.
Friedberg/Insel/Spence 4.2-4.5
Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial.
Friedberg/Insel/Spence 4.1, 5.1
Lecture 21: Eigenvalues and eigenvectors (continued). Diagonalization.
Friedberg/Insel/Spence 5.1-5.2
Lecture 22: Diagonalization (continued). Matrix polynomials.
Friedberg/Insel/Spence 5.2
Lecture 23: Diagonalization (continued). The Cayley-Hamilton theorem.
Friedberg/Insel/Spence 5.2, 5.4
Lecture 24: Multiple eigenvalues. Invariant subspaces. Markov chains.
Friedberg/Insel/Spence 5.2-5.4
Lecture 25: Markov chains (continued). The Cayley-Hamilton theorem (continued).
Friedberg/Insel/Spence 5.3-5.4
Lecture 26: Review for Test 2.
Friedberg/Insel/Spence 3.1-3.4, 4.1-4.5, 5.1-5.4
Lecture 27: Norms and inner products.
Friedberg/Insel/Spence 6.1
Lecture 28: Inner product spaces.
Friedberg/Insel/Spence 6.1
Lecture 29: Orthogonal sets.
Friedberg/Insel/Spence 6.2
Lecture 30: The Gram-Schmidt process. Orthogonal complement.
Friedberg/Insel/Spence 6.2
Lecture 31: Dual space. Adjoint operator.
Friedberg/Insel/Spence 2.6, 6.3
Lecture 32: Adjoint operator (continued). Normal operators.
Friedberg/Insel/Spence 6.3-6.4
Lecture 33: Diagonalization of normal operators.
Friedberg/Insel/Spence 6.4
Lecture 34: Unitary operators. Orthogonal matrices.
Friedberg/Insel/Spence 6.4-6.5
Lecture 35: Orthogonal matrices (continued). Rigid motions. Rotations in space.
Friedberg/Insel/Spence 6.5, 6.11
Lecture 36: Operator of orthogonal projection.
Friedberg/Insel/Spence 6.6
Lecture 37: Jordan blocks. Jordan canonical form.
Friedberg/Insel/Spence 7.1-7.2
Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
Friedberg/Insel/Spence 7.1-7.2
Lecture 39: Review for the final exam.
Friedberg/Insel/Spence 1.1-1.7, 2.1-2.5, 3.1-3.4, 4.1-4.5, 5.1-5.4, 6.1-6.6, 6.11, 7.1-7.2