Math E21b – Spring 2019 – Calendar of topics and HW assignments
Last updated
Monday, May 27, 2019 11:53 AM
Date  Topics  Text sections and homework assignments [Solutions] (Check back after class each week for possible changes) Underlined problems indicate those selected for grading. 

Thurs, Jan 31 (Class #1) 
Algebra and geometry of lines, planes; solving simultaneously m equations in n unknowns; consistent vs. inconsistent systems; row reduction and row operations; reduced row echelon form (RREF); parameterization of solutions; rank of a matrix; product of a matrix and a vector; vector and matrix forms of systems of linear equations. [Multivariable Calculus Notes #1 for vector references] 


Thurs, Feb 7 (Class #2) 
Vector and matrix forms of systems of linear equations; linear transformations from R^{m} to R^{n} defined by matrices; geometric meaning of linearity; domain and codomain; meaning of the columns of a matrix; rotations and dilations; shears; projections and reflections. Inverse of a linear transformation and method for finding the inverse of a (square) matrix.
Supplement on the dot product and orthogonal projection 


Thurs, Feb 14 (Class #3) 
Inverse of a matrix; matrix algebra; associativity and composition of linear functions; image and kernel of a linear transformation; linear combinations and the span of a set of vectors; subspaces; linear independence; basis of a subspace. Note: You may want to practice entering matrices on a calculator and performing matrix algebra with the calculator. 


Thurs, Feb 21 (Class #4) 
Test for linear independence, basis and dimension of a subspace, proof that dimension is welldefined; bases for kernels and images; RankNullity Theorem; coordinates of a vector relative to a basis; matrix of a linear transformation relative to an alternate basis. 


Thurs, Feb 28 (Class #5) 
Matrix of a linear transformation relative to an alternate basis, applications to linear transformations defined geometrically. General linear spaces, examples  continuous functions, differentiable functions, polynomials of degree less than or equal to n, the linear space of m by n matrices; complex numbers as a (real) 2dimensional linear space. Subspaces of a linear space; basis and dimension; coordinates relative to a basis. General linear transformations and their matrices relative to a basis or bases; kernel, image, isomorphisms. Practice Exam #1 Solutions 


Thurs, Mar 7 (Class #6) 
Orthogonality (perpendicularity) of vectors in R^{n}; length (norm) of a vector, unit vectors; CauchySchwartz inequality; orthogonal complements and method for finding them; introduction to orthogonal projections (see text and Lecture Notes for additional details). Exam #1 Solutions 


Mon, Mar 14 (Class #7) 
Orthogonal projections; orthonormal basis; angle between two vectors; GramSchmidt orthogonalization process; QR factorization. Orthogonal transformations and orthogonal matrices. Leastsquares approximation, normal equation; datafitting. 


Thurs, Mar 21  No Class  Harvard Spring Break  
Thurs, Mar 28 (Class #8) 
Determinant of a (square) matrix, patterns and permutations; Laplace expansion; multilinearity and the effect of the row operations on the value of the determinant; determinant criterion for invertibility of a matrix; kvolumes; determinant as an expansion factor; Cramer's Rule and formula for finding A^{1} (minors, cofactors, and the classical adjoint). 


Thurs, Apr 4 (Class #9) 
Summary of facts about determinants and applications. Discrete (linear) dynamical systems, iteration of a matrix, trajectories and phase portraits; eigenvectors and eigenvalues of a (square) matrix; characteristic polynomial; algebraic and geometric multiplicities; diagonalization and the existence of a basis of eigenvectors; powers of a matrix. A different take on determinants by Sheldon Axler: 


Thurs, Apr 11 (Class #10) 
Diagonalization and the existence of a basis of eigenvectors; powers of a matrix; trace and determinant; repeated eigenvalues; complex eigenvalues; review of facts about complex numbers; rotationdilation matrices and complex eigenvalues. 


Thurs, Apr 18 (Class #11) 
Rotationdilation matrices and complex eigenvalues; eigenvalues and stability of a discrete linear dynamical system (phase portraits). Examples of complex and repeated eigenvalue cases. Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis. Practice Exam #2 Solutions 


Thurs, Apr 25 (Class #12) 
Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis. Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses and hyperbolas. Exam 2 Solutions The exam will cover topics from Chapers 47 of the text. 


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.]  
Thurs, May 2 (Class #13) 
Systems of linear differential equations and their solutions  distinct real eigenvalue case, complex eigenvalue case. First order linear systems of differential equations and evolution matrices Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system Matrix Methods for Solving Systems of 1st Order Linear Differential Equations 


Thurs, May 9 (Class #14) 
Systems of linear differential equations and their solutions  complex eigenvalue examples, repeated eigenvalue case, and The Big Picture. Nonlinear systems and linearization around equilibria. Matrix Methods for Solving Systems of 
HW #14: Read sections 9.1 and 9.2 and the supplement on nonlinear systems and linearization. There are five problems in the nonlinear supplement and solutions are posted for those problems. Practice Final Exam Solutions 

Thurs, May 16  Twohour FINAL EXAM from 8:00pm to 10:00pm in Emerson 105 (No alternate exam dates or times will be permitted except via approved petition through the Harvard Extension School office) 