Math 31AH

• Office: APM 6-101
• Office hour: Tuesdays 4-5pm, Wednesdays 3:15-4:15. I am also availabe for questions after lecture.
• Email: doprea at math dot ucsd dot edu

• Discussion: Thursday 10-10:50AM or Thursday 11-11:50, APM 7-421
• Office: APM 6-436
• Office Hours: Mondays 1-3pm and Thursdays 2-4pm.
• Email: c1woods at ucsd dot edu

This course is the first in the Mathematics Department's Honors sequence. It covers basic linear algebra: vectors and matrices, solution of systems of linear equations, geometry of vector spaces, linear transformations, eigenvalues and eigenvectors.

The three courses in the Honors sequence, Math 31ABC, cover essentially the same material as do Math 20F, 20C, and 20E, respectively, but at a more sophisticated conceptual level. The Honors sequence emphasizes proofs, so students completing it will be exempt from taking Math 109 (Mathematical Reasoning). The Honors sequence is intended for mathematics majors and prospective mathematics majors (although others are very welcome) and gives a much better view of what upper-division mathematics is like than does the standard Math 20 sequence.

Reading 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 problems will be assigned on the course homework page. There will be 7 problem sets, due certain Fridays at 4:30PM in the TA's mailbox. The tentative dates for the homeworks are Oct 4, Oct 11, Oct 18, Nov 1, Nov 8, Nov 15, Dec 4.

You may work together with your classmates on your homework and/or ask the TA (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.

The final examination will be held on Thursday, December 12, 11:30-2:30, Peterson 102. 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.

• Friday, September 27: First lecture
• Monday, October 21: Midterm I
• Monday, November 11: Veterans' Day. No class
• Monday, November 18: Midterm II
• Thursday-Friday, November 28-29: Thanksgiving. No class.
• Friday, December 6: Last Lecture
• Thursday, December 12: FINAL EXAM, 11:30AM-2:30PM pm

• Preparation for Midterm 1:
• Practice Midterm 1 (Fall 2010) and Solutions
• Midterm 1 and Solutions
• Preparation for Midterm 2:
• Practice Midterm 2 (Fall 2010) and Solutions
• Another Practice Midterm 2
• Midterm 2 and Solutions
• Preparation for Final Exam:
• Practice Final (Fall 2010) and Solutions
• Additional Practice Problems and Solutions
• Final Exam and Solutions

• Lecture 1: Introduction. Vectors. Addition. Scalar multiplication. Vector subspaces.
• Lecture 2: Matrices. Operations with matrices: addition, scalar multiplication, multiplication, properties, matrix-vector product, inverses, transpose.
• Lecture 3: Dot product. Angle between vectors. The Cauchy-Schwartz inequality. The triangle inequality. Length of matrices.
• Lecture 4: Cross product and properties. Solving systems of linear equations. Row operations. Row reduced echelon form.
• Lecture 5: Solving systems using row reduction. Pivot and free variables. Uniqueness and existence of solutions. Existence of the row-reduced echelon form. Mathematical induction.
• Lecture 6: Linear independence. Span. Basis. Dimension of a subspace.
• Lecture 7: Inverses. Finding inverses by row reduction. Elemenatary matrices.
• Lecture 8: Null space. Basis for the null space. Dimension of the null space is given by the number of free variables. Determining if the columns of a matrix are linearly dependent.
• Lecture 9: Column space. Finding the column space by row-reducing the augumented matrix. Basis for column space is given by the pivot columns. Rank. Rank-nullity theorem.
• Lecture 10: Row space and row-rank. Row-rank equals column-rank. Inverses and column space/null space. Left inverses and right inverses. Midterm review.
• Lecture 11: Linear transformations. Matrix of a linear transformation. Geometric examples: projections, reflections, rotations.
• Lecture 12: Injective, surjective, bijective maps. Image and pre-image. Reinterpretation of these notions for linear transformations. Composition of linear transformations and matrix multiplication.
• Lecture 13: Determinants. Multilinear, anti-symmetric, normalized functions. Expansion along columns. Effect of column operations on determinants.
• Lecture 14: Uniquness of determinants as multi-linear, anti-symmetric, normalized functions. Invertible matrices have non-zero determinants. Determinants distributes over matrix products.
• Lecture 15: Connection between determinants and volumes. Existance of determinants as multilinear, antisymmetric, normalized functions. More on inductive proofs.
• Lecture 16: Every invertible matrix is product of elementary matrices. Determinant of the transpose. Determinants and row operations. Permutations and signature. Permutation matrices.
• Lecture 17: Determinants and permutations. Orthogonal complements and properties.
• Lecture 18: Connections between the four subspaces of a matrix. Matrix of the projection onto the column space. Connection with left inverses.
• Lecture 19: Projections onto row space, column space, null space, left null space. Least square solutions and connection with the left inverse. Best fit line.
• Lecture 20: Orthogonal bases, orthogonal matrices. Formulas for projections in the presence of an orthonormal basis. Gram-Schmidt process.
• Lecture 21: Systems of coordinates. Change of basis matrix. Matrix of a linear transformation with respect to an arbitrary basis. Example: reflection in a plane in R^3.
• Lecture 22: Similar matrices. Matrices of linear transformation in different bases are similar. Diagonalizable linear transformations. Eigenvalues, eigenspaces, characteristic polynomial.
• Lecture 23: Examples of eigenvalues and eigenspaces. Sum and product of eigenvalues in terms of trace and determinant. Diagonalizable matrices in terms of geometric/algebraic multiplicities.
• Lecture 24: Similar matrices have the same characteristic polynomial. Computing powers of diagonalizable matrices. Cayley-Hamilton theorem. Symmetric matrices and the spectral theorem.
• Lecture 25: More on the spectral theorem and examples. Quadratic forms and symmetric matrices.
• Lecture 26: Signature of quadratic forms and classification in terms of eigenvalues. Sylvester's criterion.