Lecture Summaries and Homework Assignments

Lecture 25, Tuesday, May 5:  Read Section 6.6. Do Section 6.6 problems 1, 2, 6, 9, 11.
Linear models, including one from auto insurance. Brief review of the semester.
Lecture 24, Thursday, April 30: 
Least-squares approximation and examples.
Lecture 23, Tuesday, April 28:  Read Sections 6.4, 6.5, and 6.6. Do Section 6.4 problem 10 and Section 6.5 problems 2, 6, all to be handed in on Tuesday, May 5 in class. Also do problems 1, 3, 7, 9, 14 from Section 6.5, but do not turn any of these in.
Orthogonal projection via the transpose. Least-squares approximation to solution of an inhomogeneous linear system: general method and examples.
Lecture 22, Thursday, April 23:  Do Section 6.1 problems 1, 2, 3, 5, 9, 15, 16, 17, 18, 26; and Section 6.2 problems 1, 2, 5; none of these should be handed in. Also, do Section 6.2 problems 10, 14 and Section 6.3 problems 2, 10, 12; these five should be handed in on Thursday, April 30, in class.
Orthogonal bases and orthogonal projection onto a subspace. How to compute it.
Lecture 21, Thursday, April 16:  Read Sections 6.1, 6.2, 6.3. Problems to be assigned.
Inner product (dot product), length, orthogonality, orthogonal projection along a vector, orthogonal sets and orthogonal projection onto a subspace.
Lecture 20, Tuesday, April 14:  Read Section 5.7. Do Section 5.7 problems 1, 2, 9, 10 (but for 9, 10, don't bother to describe the shapes of trajectories). Study for the midterm.
Applications of complex eigenvalues to solving some particular differential equations.
Lecture 19, Thursday, April 9:   Read Section 5.5. Do Section 1.4 problems 17, 19. Do Section 1.7 problems 1, 5. Do Section 3.2 problem 22. Do Section 5.2 problems 7, 11, 19. Do Section 5.3 problems 11, 13, 17, 18. Do Section 5.4 problems 2, 4. Of these, hand in the following on Thursday, April 16: Section 1.4 problem 17; Section 3.2 problem 22; Section 5.2 problem 11; Section 5.3 problem 18; Section 5.4 problem 2.

You must show your work on each problem to get any credit whatsoever on that problem.

Diagonalizing a matrix: examples. What can go wrong: a 2x2 matrix with only one eigenvalue, and having only a 1-dimensional space of eigenvectors for that eigenvalue---so it can't be diagonalized! Even worse: rotations. Complex numbers and how they're used to understand such a matrix (beginnings).
Lecture 18, Tuesday, April 7:  Read Sections 5.2, 5.3, 5.4 again. Catch up on homework and go over the quiz.
Eigenvalues and eigenvectors. How to find eigenvalues using the characteristic polynomial. How to find eigenvectors for a given eigenvalue. How to diagonalize a matrix.
Lecture 17, Tuesday, March 31:  Read Sections 5.1 through 5.4. Do Section 5.1 problems 1, 3, 9, 13; Section 5.2 problems 1, 5, 13, 17; Section 5.3 problems 1, 3. Also do Section 5.1 problems 8 and 10, Section 5.2 problems 10 and 18, and Section 5.3 problem 4; these last five problems should be handed in on Thursday, April 9.
Eigenvalues and eigenvectors. Linear independence of eigenvectors with distinct eigenvalues. Determinants and the characteristic polynomial. Beginnings of diagonalization.
Lecture 16, Thursday, March 19:  Do Section 3.1 problems 3, 9, 13, 25, 27; Section 3.2 problems 21, 25, 33, 34; Section 3.3 problems 1, 3, 11.
Chapter 3: all about determinants! Mainly, definition/formula, relation to invertibility, using it to solve equations, finding matrix inverses using Cramer's rule.
Lecture 15, Tuesday, March 17:  Read Chapter 3. Do Section 2.2 problems 31, 32, 33; Section 4.9 problems 1, 3, 9, 11, 17 (not to be handed in). Also do Section 4.9 problems 4, 16, 18 and Section 3.1 problems 2 and 10, all to be handed in on Thursday, April 2.
Absorbing Markov chains; example from squash.
Lecture 14, Thursday, March 12:  Make sure you can do all the problems on the midterm. Read Section 4.9.
Example of diagonalization of a matrix by a good choice of basis. Definition and first properties of Markov chains. Examples.
Lecture 13, Tuesday, March 10:  Read Sections 4.6 and 4.7. Do Section 4.5 problem 21 and Section 4.7 problems 1, 5, 7, 9, 19 (not to hand in). Do Section 4.5 problem 20 and Section 4.6 problem 20 and Section 4.7 problems 2, 8, 14, all to be handed in on Thursday, March 12 in class.
Isomorphisms, coordinate transformations. Change of basis in an example. Systematic treatment of the matrix of a linear transformation with respect to a given basis. What happens to that matrix when you change the basis. How to compute the matrix in the new basis via an example.
Lecture 12, Tuesday, March 3:   Read Sections 4.6 and 4.7. See the next lecture for homework.
Bases and coordinates. Coordinate mappings.
Lecture 11, Thursday, February 26:  Read Sections 4.4 and 4.5. Do Section 4.2 problems 1, 5, 7, 15, 25, 26, 27, 31, 33, 34. Do Section 4.3 problems 9, 13, 15, 19, 25.
Vector spaces and subspaces. Linear transformations, their kernels and images. Linear independence and bases. Examples: the vector space of all polynomials in a variable x, differentiation as a linear transformation. First steps: bases tell us coordinates.
Lecture 10, Tuesday, February 24:Read Sections 4.1, 4.2, 4.3. Do Section 2.9 problems 1, 3, 5, 9, 11, 13, 19, 21, 25. Do Section 4.1 problems 1, 3, 7, 13, 19. [None of these to be handed in: we won't have homework to be handed in next week.]
Bases and dimension. Dimension of the null space of a matrix. Rank of a matrix and how to compute it. Definition of a vector space and some examples.
Lecture 9, Thursday, February 19: Read Sections 2.8 and 2.9. Do Section 2.6 problems 9, 10, 11 and Section 2.8 problems 8, 24, all to be handed in (due Thursday, February 26 in class). Also do Section 2.8 problems 1, 2, 3, 4, 5, 7, 17, 21, 31, none of these to be handed in.
Leontiev input-output models. Subspaces. Column space, null space. Examples and calculations.
Lecture 8, Tuesday, February 17: Read Section 2.5. Do Section 2.3 problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 27, 33. Read the "numerical notes" box at the bottom of page 131 and do problems 42 and 44.
Elementary matrices and inverses. Algorithm for inverting a matrix. Characterization of invertible matrices.
Lecture 7, Thursday, February 12: Read Sections 2.1 through 2.4. Do Section 2.1 problems 1, 3, 5, 9, 13, 17, 21, 23; and Section 2.2 problems 1, 3, 9, 13, 21, none of these to be handed in. Do Section 2.1 problems 16, 24, 26 and Section 2.2 problems 6, 16 to be handed in (due Thursday, February 19 in class).
Matrix operations: sum, scalar multiple, product. Interpretation of product via composition of linear transformations. How to compute the product. When can a linear transformation have an inverse? Dimension 2. Discussion of the geometry behind existence of an inverse in dimension 3.
Lecture 6, Thursday, February 5: Read Sections 1.8 and 1.9. Do Section 1.6 problem 4, Section 1.7 problems 26 and 28, Section 1.8 problem 17 (justify your answer!), and Section 1.9 problem 10, all to be handed in (due Thursday, February 12 in class). Also do Section 1.8 problems 3, 11, 24, 28, 29; and Section 1.9 problems 3, 5, 15, 19, 36.
Linear transformations. The linear transformation associated to a matrix. The matrix of a linear transformation. Differentiation as a linear transformation.

ANNOUNCEMENT:Next week (February 9-13) my office hour will be on Monday from 4-5 p.m., NOT Wednesday.
Lecture 5, Tuesday, February 3:   Read Sections 1.7 and 1.6. Do Section 1.5 problems 7, 9, 11, 13, 17, 23. Do Section 1.7 problems 1, 3, 7, 9, 15, 23. Do Section 1.6 problems 1, 3.
Linear independence and dependence. How to tell whether a set is linearly independent. The Leontief exchange model (section 1.6). We also worked a bit on these exercises to build your linear algebra muscles.
Lecture 4, Thursday, January 29:  Do Section 1.4 problems 17, 18, 30 and Section 1.5 problems 12, 28, all to be handed in (due Thursday, February 5 in class). Solutions of homogeneous and inhomogeneous systems, including parametric description.
Lecture 3, Tuesday, January 27:  Read Sections 1.3, 1.4 and 1.5. Do Section 1.3 problems 1, 2, 3, 5, 6, 9, 10, 11, 13, 15, 19, 22, 23, 28. Do Section 1.4 problems 1, 3, 4, 5, 7, 11.
Vectors and their geometric meaning. Vector equations and matrix equations (sections 1.3 and 1.4). Linear combinations; span of a collection of vectors. How to solve a vector or matrix equation by using row reduction.
Lecture 2, Thursday, January 22:  Do Section 1.1 problem 26 and Section 1.2 problems 11, 12, 20, 22 (be sure to justify your answers!) to be handed in (due Thursday, January 29 in class). See this page for a discussion of the rules for homework assignments.
Also do Section 1.2 problems 1, 2, 3, 4, 13, 15, 16, 26 but do not hand them in.
Row echelon form and reduced row echelon form; row reduction; how to tell if a system of linear equations is consistent; how to tell if a system of linear equations that is consistent has a unique solution.
Lecture 1, Tuesday, January 20:  Read Sections 1.1, 1.2, and 1.3. Do Section 1.1 problems 1, 3, 7, 11, 13, 23, 24, 26, 29, 31, 33 (none of these will be handed in).
What is this course about? Systems of linear equations; geometric interpretation; matrix associated to a system of linear equations; elementary row operations; beginnings of row reduction.
Main course page.