Math S21b  Calendar of topics and HW assignments  Summer 2019
Last updated
Wednesday, February 27, 2019 2:35 PM.
[This Calendar will be updated as the course progresses. Check back frequently.]
Date  Topics  Homework assignments  
Mon, June 24 week #1 
Algebra and geometry of lines, planes; solving simultaneously m equations in n unknowns; row reduction and row operations; reduced row echelon form; parametric representation of solutions, consistent vs. inconsistent systems, rank of a matrix. [June/July 2011 AMS Notices article: Mathematicians of Gaussian Elimination] [Online RREF Calculator] For those who may need a refresher in vector operations, see Appendix A at the back of the Bretscher text. You may want to learn how to enter a matrix into a matrixcapable calculator and how to use the “rref” function to do row reduction. 


Wed, June 26 
Product of matrix and vector, matrix form of a linear system. Linear transformation defined by a matrix; linearity property; meaning of the columns of a matrix. Identity matrix, dilation (scaling) matrix. Constructing matrices of common linear transformations in geometry  rotations, dilations, projections, reflections. Supplement on the dot product and orthogonal projection 


Fri, June 28 
Algebraic and geometric properties of the dot product; projections and reflections. Invertible linear transformations and algorithm for finding the inverse of a matrix (when it exists). Matrix algebra, composition of linear functions and matrix products. 

Mon, July 1 week #2 
Subspaces of R^{n}, span of a collection of vectors; kernel and image of a linear transformation. Linear dependence and linear independence. Basis and dimension of a subspace; RankNullity Theorem. 


Wed, July 3  
Fri, July 5 
Coordinates of a vector relative to a basis; examples. Matrix of a linear transformation relative to an alternate basis; interpretation of the columns of a matrix relative to a given basis; similarity of matrices. Supplement on Representation of Functions in Different Coordinates 

Mon, July 8 week #3 
Introduction to general linear spaces (vector spaces), e.g. function spaces and families of matrices. Linear transformations from one vector space to another, isomorphisms, coordinates relative to a basis for a linear space or subspace. Image, kernel, rank, nullity of general linear transformations. Introduction to orthogonality, orthonormal bases, orthogonal complement of a subspace of R^{n}. Practice Exam #1 Solutions 


Wed, July 10 
Introduction to orthogonality, orthonormal bases, orthogonal complement of a subspace of R^{n}. Finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis. Orthogonal matrices; facts about transposes. Midterm Exam #1 (covering Chapters 13 of text) 


Fri, July 12 
GramSchmidt process and QR factorization. Orthogonal transformations; orthogonal matrices. Method of Least Squares for approximate solutions to a linear system. Regression lines, data fitting, alternate method for finding matrix for orthogonal projection. Lecture #9 Notes (includes some topics to be covered on Monday) 

Mon, July 15 week #4 
Inner product spaces; Fourier approximation, Fourier coefficients, Fourier series, and applications to infinite series. Lecture #10 Notes (includes some topics covered Wednesday) 


Wed, July 17 
Last details of inner product spaces. Determinant of a square matrix, permutations, Laplace expansion, multilinearity, effect of row operations on the value of the determinant; matrix A invertible if and only if det(A) nonzero. Lecture #11 Notes (includes some topics covered Friday) 

Fri, July 19 
Determinants, continued: det(AB) = det(A)det(B) and its corollaries; determinant of a linear transformation. Use of determinants to find area, volume, kvolume; determinant as an expansion factor; Cramer's Rule and a not too useful formula for calculating the inverse of a matrix (see text). Eigenvalues and eigenvectors of a (square) matrix; characteristic polynomial; finding eigenvalues and eigenvectors. Algebraic multiplicity (AM) vs. geometric multiplicity (GM) of an eigenvalue, diagonalization and criteria for the existence of a basis of eigenvectors. Lecture #12 Notes (includes some topics to be covered on Monday) 
Practice Exam #2 Solutions 

Mon, July 22 week #5 
Distinct eigenvalues yield linearly independent eigenvectors; diagonalizability. Discrete linear dynamical systems, powers of a matrix, and Markov matrix example. Phase portraits; similar matrices have the same characteristic polynomials, same eigenvalues, same algebraic and geometric multiplicities. Obstructions to diagonalizability (GM < AM, complex eigenvalues). Trace and determinant in terms of eigenvalues; characteristic polynomial and eigenvalues of a linear transformation. Lecture #13 Notes (some topics to be covered Wednesday) 

Wed, July 24 
Review of complex numbers; complex eigenvalues and invariant (rotationdilation) subspaces. Midterm Exam #2 (covering Chapters 46 and 7.17.4) 


Fri, July 26 
Dealing with repeated eigenvalues where GM < AM. Stability of a discrete linear dynamical system in terms of the modulus of the eigenvalues. Final details on eigenvalues and the Jordan canonical form of a matrix. Spectral Theorem and orthogonal diagonalizability of symmetric matrices. Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses, hyperbolas, ellipsoids and hyperboloids; 2nd derivative test for functions of several variables in terms of eigenvalues. Supplement on quadratic forms, critical points, principal axes, and the 2nd derivative test 

Here's a website that has a good javabased tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.] 


Mon, July 29 week #6 
Continuation of quadratic forms and Principal Axes Theorem. Vector fields; systems of linear ordinary differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix. Complex eigenvalue case; reduction of order; Hooke's Law and oscillations. Lecture #16 Notes (some topics to be covered Wednesday) 

Wed, July 31 
Vector fields and continuous dynamical systems, continued  complex eigenvalues, damped spring example, repeated eigenvalues, combination problems; stability, phaseplane analysis, analytic solutions. Linear differential operators and solutions to homogeneous and inhomogeneous linear differential equations, eigenfunctions, characteristic polynomial; kernel and image of a linear differential operator. Supplement on Matrix Methods for Solving 
Practice Final Exam Solutions to Practice Final Exam 

Fri, Aug 2 
Linear operators, continued. Nonlinear differential equations; nullclines, equilibria. Equilibrium analysis using Jacobian matrices; phaseplane analysis. 

Mon, Aug 5 
Final Exam Review at 9:30am in Harvard Hall 201  
Fri, Aug 9 
FINAL EXAM (8:30am  11:30am) in Harvard Hall 201 (our regular lecture hall) There will be no alternate arrangements for this exam except for those with certified disabilities. 