Meeting TimeMon., Wed., Fri., 2PM - 2:50PM
Location CENTER 113
Instructor Dragos Oprea
Christopher Briggs Daniel Minsky
Textbook Gilbert Strang, Applied Linear Algebra, 4th edition.
Required. Available at the bookstore and on reserve in the library.
The grade is computed as the following weighed average:
  • Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.
Math 102 can be thought of as a sequel to Math 20F. The topics covered will include row reduction, LU decomposition, the four spaces associated to a matrix, projections, Gram-Schmidt, QR decompositions, least squares, eigenvalue problems, difference equations, abstract vector spaces, linear transformations, abstract inner products, Hermitian matrices, singular value decomposition, pseudoinverses.
Prerequisites Math 20F or Math 31AH.
ReadingsReading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.
Homework Homework problems will be assigned on the course homework page. There will be 7 problem sets, due certain Wednesdays at 4:30pm in the TA's mailbox. You may work together with your classmates on your homework and/or ask the TA's (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.
There will be two midterm exams given in class. The dates are January 28 and February 27. There will be no makeup exams.
The final examination will be held on Monday, March 18, 3-6 PM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.
Announcements and Dates
  • Monday, January 7: First lecture
  • Monday, January 21: MLK Day, no class.
  • Wednesday, January 28: Midterm I
  • Monday, February 18: Presidents' Day, no class.
  • Wednesday, February 27: Midterm II
  • Friday, March 15: Last Lecture
  • Monday, March 18: FINAL EXAM, 3-6PM.
Lecture Summaries
  • Lecture 1: Introduction. Elimination. Row operations. Row reduction. Row echelon form. Row reduced echelon form.
  • Lecture 2: Effect of row operations on a matrix. Elementary matrices. LU decomposition. Existence and uniqueness. LDU and PLU decomposition.
  • Lecture 3: LDL decomposition of symmetric matrices. Using LU decompositions to solve systems. Solving systems via inverses. Computing inverses via row reduction. Homogeneous vs. inhomogeneous systems.
  • Lecture 4: Abstract vector spaces and subspaces. Examples: spaces of functions, spaces of polynomials, spaces of matrices, spaces of sequences, solutions to differential equations, and others.
  • Lecture 5: Linear independence, span, bases, dimension for abstract vector spaces. Properties and examples.
  • Lecture 6: The four subspaces associated to a matrix and their dimensions. Column spaces, null space, row space, left null space. Rank and nullity. Finding bases for the four subspaces.
  • Lecture 7: Existence of left and right inverses. How to compute left and right inverses. Examples.
  • Lecture 8: Abstract linear transformations and examples. Rotations, projections, reflections, differentiation, integration. Matrix of a linear transformation in a given basis.
  • Lecture 9: Matrix of an abstract linear transformation. Reading the four subspaces of an abstract linear transformation from matrix representation. Midterm review.
  • Lecture 10: Dot product and orthogonality. Properties of dot product. Orthogonal complements and properties. Orthogonality and the four subspaces of a matrix.
  • Lecture 11: Matrix of projections and the left inverse. Least squares solutions. Weighted least squares. Best fit line. Examples. How does your GPS work?
  • Lecture 12: Orthonormal bases. Projections when an orthonormal basis is given. Orthogonal matrices. Gram-Schmidt process. Examples.
  • Lecture 13: Gram-Schmidt process and QR decomposition. Existence and uniqueness. Connections with left inverses and least squares. Examples.
  • Lecture 14: Abstract inner product spaces and examples. Abstract Gramm-Schmidt for spaces of functions. Legendre polynomials. Fourier coefficients.
  • Lecture 15: Determinants. Definition and properties. Computing determinants via row reduction, via cofactors, via permutation matrices. Invertible matrices.
  • Lecture 16: Applications of determinants: finding inverses via cofactors, Cramer's rule, volumes.
  • Lecture 17: Eigenvalues and eigenvectors. Examples. Characteristic polynomials. 2x2 matrices. Multiplicity of eigenvalues. Relation to trace and determinants.
  • Lecture 18: Similar matrices, diagonalizable matrices. Simultaneously diagonalizable matrices. Similar matrices share the same characteristic polynomial, eigenvalues, trace, determinant.
  • Lecture 19: Powers of matrices and exponentials via diagonalization. Difference and differential equations. Midterm review.
  • Lecture 20: Fibonacci sequences. Stability of difference and differential equations. Markov matrices. Perron-Frobenius. Google page rank. Other examples.
  • Lecture 21: Symmetric matrices and diagonalization. Skew symmetric matrices and stability of differential equations. Complex vectors. Hermitian product and properties. Hermitian transpose.
  • Lecture 22: Hermitian matrices, skew Hermitian matrices, unitary matrices, normal matrices. Properties of their eigenvalues and eigenspaces. Examples.
  • Lecture 23: Unitary similarity. Unitary triangularization. Unitary diagonalization. Examples. Spectral theorem for Hermitian and symmetric matrices.
  • Lecture 24: Quadratic forms and their definiteness. Criteria for definiteness. Positive decomposition of symmetric matrices. Writing quadratic forms as sums of squares.
  • Lecture 25: Singular value decomposition. Left and right singular vectors. Singular values. Calculating the singular value decomposition and examples.
  • Lecture 26: Applications of singular value decomposition. Reading off the 4 subspaces of a matrix, projections onto the four subspaces of a matrix, pseudoinverse, least squares solutions of minimal length.
  • Lecture 27: More examples and course review.