### Differential Equations Math 2270-004 Spring 2018 Lecture Page

Department of Mathematics
College of Science
University of Utah

Lecture notes for each week should be posted by Thursday at 3:00 p.m. on the preceding week. Printing for Math classes is free in the Rushing Student Center, in the basement of LCB. It is recommended that you use these notes in conjunction with attending class. After each class I will post the filled-in versions for that day. At the end of the week I'll post the entire week's filled-in notes.

Week 1: January 8-12 Sections 1.1-1.3
week1.pdf Notes that include outline for entire week
The daily post-notes from this week are below:
jan8.pdf syllabus, 1.1. linear systems of equations
jan9.pdf 1.2 Gaussian elimination
jan10.pdf 1.2 reduced row echelon form, consistent and inconsistent systems.
jan12.pdf 1.2, intro to 1.3. reduced row echelon form, intro to vectors.
week1_post.pdf all material we covered in week 1

Week 2: January 16-19 Sections 1.3-1.5
week2.pdf Notes that include outline for entire week
jan16.pdf 1.3 vector equations
jan17.pdf 1.4 vector equations and matrix equations are systems of linear equations.
jan19.pdf 1.5 solution sets of linear systems
week2_post.pdf all material we covered in week 2

Week 3: January 22-26 Sections 1.6-1.8
week3.pdf Notes that include outline for entire week
jan22.pdf review of 1.1-1.5 concepts and conclusions
jan23.pdf 1.6 applications, and introduction to 1.7 linear dependence/independence
jan24.pdf 1.7 linear dependencies correspond to solutions to homogeneous matrix equations; rref of a matrix encodes column dependencies.
jan26.pdf 1.8 Matrix transformations are linear transformations, examples.
week3_post.pdf all material we covered in week 3

Week 4: January 29 - February 2 Sections 1.8-1.9, 2.1-2.2
week4.pdf Notes that include outline for entire week
jan29.pdf 1.9 linear transformations from R2 to R2
jan30.pdf 1.9 continued
jan31.pdf 1.9, 2.1 matrix multiplication arises from composition of linear transformations.
feb2.pdf 2.1 matrix algebra
week4_post.pdf all material we covered in week 4

Week 5: February 5-9 Sections 2.1-2.3
week5.pdf Notes that include outline for entire week
feb5.pdf 2.1-2.2 matrix algebra and introduction to matrix inverses
feb6.pdf 2.2-2.3 matrix inverses
feb7.pdf 2.3 big theorem relating key concepts, and giving a long list of conditions equivalent to a matrix having an inverse.
week5_post.pdf all material we covered in week 5

Week 6: Feb 12-16 Sections 2.3, 3.1-3.3
week6.pdf Notes that include outline for entire week
feb12.pdf 2.3 matrix inverses via products of elementary matrices
feb13.pdf 3.1 determinants
feb14.pdf 3.2 determinant properties
feb16.pdf 3.3 formula for inverse matrices, Friday fft.
week6_post.pdf all material we covered in week 6

Week 7: Feb 20-23 Sections 4.1-4.3
week7.pdf Notes that include outline for entire week
feb20.pdf 4.1 vector spaces and subspaces
feb21.pdf 4.1-4.2 Linear combinations, span, linear dependence/independence in vector spaces. Subspaces as spans of collections of vectors.
feb23.pdf 4.2-4.3 nullspace, column space, basis.
week7_post.pdf all material we covered in week 7

Week 8: Feb 26 - Mar 2 Sections 4.2-4.6
week8.pdf Notes that include outline for entire week
feb26.pdf 4.2-4.3 good bases for subspaces of Rn.
feb27.pdf 4.2 general linear transformation between vector spaces, kernel and range subspaces.
feb28.pdf 4.4-4.5 bases for vector spaces yield coordinate systems.
mar2.pdf 4.4-4.5 vector space isomorphisms and coordinates, change of coordinates matrix introduction.
week8_post.pdf all material we covered in week 8

Week 9: Mar 5-9 Sections 4.5-4.6, 5.1-5.2
week9.pdf Notes that include outline for entire week
mar5.pdf 4.5-4.6 vector space theorems; 4 subspaces associated with matrices.
mar6.pdf 4.5-4.6 completed
mar7.pdf 4.7 transition matrices to change coordinates in a vector space.
mar9.pdf 5.1-5.2 introduction to eigenvectors and eigenvalues
week9_post.pdf all material we covered in week 9

Week 10: Mar 12-14 Sections 5.2-5.4
week10.pdf Notes that include outline for entire week
mar12.pdf 5.2 using the characteristic equation to find eigenvalues and eigenspace bases.
mar13.pdf 5.3 diagonalizable and non-diagonalizable matrices
mar14.pdf 5.4 Matrix of a linear transformation T:V->W, given bases for V and W.
week10_post.pdf all material we covered in week 10

Week 11: Mar 26-30 Sections 5.4-5.6
week11.pdf Notes that include outline for entire week
mar26.pdf 5.4 Matrix of a linear transformation T:V->W: "big picture" idea that encompasses our work on coordinates, change of coordinates, diagonalization, and more.
mar27.pdf 5.5 complex eigenvalues and eigenvectors
mar28.pdf 5.6 introduction to discrete dynamical systems, and understanding them via eigenvectors and eigenvalues.
week11_post.pdf all material we covered in week 11

Week 12: Apr 2-6 Sections 6.1-6.4
week12.pdf Notes that include outline for entire week
apr2.pdf 6.1-6.2 algebra and geometry of dot product; orthogonality and projection onto a line.
apr3.pdf 6.2-6.3 angles between vectors in Rn; orthogonal complements to subspaces, four fundamental subspaces of a matrix revisited, geometrically.
apr4.pdf 6.3-6.4 orthogonal and orthonormal bases for subspaces W, and projection onto W.
apr6.pdf 6.4 The Gram-Schmidt algorithm for creating orthonormal bases.
week12_post.pdf all material we covered in week 12

Week 13: Apr 9-13 Sections 6.4-6.8
week13.pdf Notes that include outline for entire week
apr9.pdf 6.4 A=QR decomposition
apr10.pdf 6.5 Least squares solutions
apr11.pdf 6.6 best data fits for appropriate models, via least squares
apr13.pdf 6.7-6.8 inner product spaces and Fourier series
week13_post.pdf all material we covered in week 13

Week 14: Apr 16-20 Sections 6.8, 7.1-7.2
week14.pdf Notes that include outline for M-W...Friday notes to be added.
apr16.pdf 6.8 Fourier series example worked out; "show and tell" about .jpg image compression (more pages available in hard copy).
apr17.pdf 7.1-7.2 Spectral theorem and quadratic forms
apr18.pdf 7.1-7.2 diagonalizing quadratic forms; matrix multiplication via "outer product"; spectral decomposition theorem for symmetric matrices.
week14_post.pdf all material we covered in week 14, except for Prof. Alberts' lecture on statistics applications of principal component analysis.

Week 15: Apr 23-24 Section 7.2 and review
week15.pdf Notes for Monday; review notes to be added later.
apr23.pdf 7.2 multivariable second derivative test, revisited.
apr24.pdf course review notes