Introduction

Skip to main content

\(\DeclareMathOperator{\RE}{Re} \DeclareMathOperator{\IM}{Im} \DeclareMathOperator{\ess}{ess} \DeclareMathOperator{\intr}{int} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\span}{span} \DeclareMathOperator{\null}{null} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\col}{col} \DeclareMathOperator{\row}{row} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator\re{\mathrm {Re~}} \DeclareMathOperator\im{\mathrm {Im~}} \newcommand\dd{\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\hilbert}{\mathcal{H}} \newcommand{\s}{\mathcal{S}_2} \newcommand{\A}{\mathcal{A}} \newcommand\h{\mathcal{H}} \newcommand{\J}{\mathcal{J}} \newcommand{\M}{\mathcal{M}} \newcommand{\F}{\mathbb{F}} \newcommand{\N}{\mathcal{N}} \newcommand{\T}{\mathbb{T}} \newcommand{\W}{\mathcal{W}} \newcommand{\X}{\mathcal{X}} \newcommand{\D}{\mathbb{D}} \newcommand{\C}{\mathbb{C}} \newcommand{\BOP}{\mathbf{B}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\BH}{\mathbf{B}(\mathcal{H})} \newcommand{\KH}{\mathcal{K}(\mathcal{H})} \newcommand{\pick}{\mathcal{P}_2} \newcommand{\schur}{\mathcal{S}_2} \newcommand{\R}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Field}{\mathbb{F}} \newcommand{\RPlus}{\Real^{+}} \newcommand{\Polar}{\mathcal{P}_{\s}} \newcommand{\Poly}{\mathcal{P}(E)} \newcommand{\EssD}{\mathcal{D}} \newcommand{\Lop}{\mathcal{L}} \newcommand{\cc}[1]{\overline{#1}} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\seq}[1]{\left\lt#1\right>} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\essnorm}[1]{\norm{#1}_{\ess}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\ran}[1]{\operatorname{ran}#1} \newcommand{\nt}{\stackrel{\mathrm {nt}}{\to}} \newcommand{\pnt}{\xrightarrow{pnt}} \newcommand{\ip}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\ad}{^\ast} \newcommand{\inv}{^{-1}} \newcommand{\adinv}{^{\ast -1}} \newcommand{\invad}{^{-1 \ast}} \newcommand\Pick{\mathcal P} \newcommand\Ha{\mathbb{H}} \newcommand{\HH}{\Ha\times\Ha} \newcommand\Htau{\mathbb{H}(\tau)} \newcommand{\vp}{\varphi} \newcommand{\ph}{\varphi} \newcommand\al{\alpha} \newcommand\ga{\gamma} \newcommand\de{\delta} \newcommand\ep{\varepsilon} \newcommand\la{\lambda} \newcommand\up{\upsilon} \newcommand\si{\sigma} \newcommand\beq{\begin{equation}} \newcommand\ds{\displaystyle} \newcommand\eeq{\end{equation}} \newcommand\df{\stackrel{\rm def}{=}} \newcommand\ii{\mathrm i} \newcommand{\vectwo}[2] { \begin{pmatrix} #1 \\ #2 \end{pmatrix} } \newcommand{\vecthree}[3] { \begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \newcommand\blue{\color{blue}} \newcommand\black{\color{black}} \newcommand\red{\color{red}} \newcommand\nn{\nonumber} \newcommand\bbm{\begin{bmatrix}} \newcommand\ebm{\end{bmatrix}} \newcommand\bpm{\begin{pmatrix}} \newcommand\epm{\end{pmatrix}} \numberwithin{equation}{section} \newcommand\nin{\noindent} \newcommand{\nCr}[2]{\,_{#1}C_{#2}} \newcommand{\vec}[1]{{\bf #1}} \newcommand{\ps}{\displaystyle \sum_{n=0}^\infty a_n x^n} \newcommand{\psg}{\displaystyle \sum_{n=0}^\infty b_n x^n} \newcommand{\hz}{\,\mathrm{Hz}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section 1.1 Introduction

This course is the second course in a sequence that uses the language and techniques of linear algebra to study problems in differential equations. The following sections give a broad overview of ideas from calculus, differential equations, and linear algebra that will be used in the course. An overview of ideas you should have seen before might look like

Calculus
- Taylor series
- Interval of convergence and the ratio test
- Power series of basic functions
- Integrating and differentiating power series
- Improper integrals
Differential equations
- Second order constant coefficient linear differential equations
- Characteristic equations
- Reduction of order
- Harmonic motion (homogeneous and forced motion)
Linear Algebra
- Vector spaces
- Linear independence
- Basis of a vector space
- Inner products (beyond the dot product)
- Orthogonality
- Vector projections

You don't need to be an expert, but having seen and used these ideas in the past is going to make this course quite a bit more interesting and productive.