Math 287 Differential Equations and Linear Algebra
- Page ID
- 227315
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1: First Chapter
- The topics of Linear Algebra and Differential Equations are very important in all the STEM fields. This book is an ongoing process which is edited to be utilized by students at Santa Ana College as well as others. The goal is to be constantly updated year over year to adapt to the growing trends in education as well as reflect the growing trends in industry.
- 2: Systems of Equations
- Designed for week 1. Matrices and Systems of Linear Equations. Systems in matrix form; relationship between coefficient matrix invertibility and solutions to a system of linear equations using Gaussian and Gauss-Jordan elimination. Inverse matrices; nature of the solution space, row space, column space, null space, rank and nullity.
- 2.1: Systems of Equations, Geometry
- 2.2: Systems of Equations, Algebraic Procedures
- 2.3: Gaussian Elimination
- 2.4: Uniqueness of the Reduced Row-Echelon Form
- 2.5: Rank and Homogeneous Systems
- 2.6: Balancing Chemical Reactions
- 2.7: Dimensionless Variables
- 2.8: An Application to Resistor Networks
- 2.9: Exercises
- 3: Matrices
- Designed for week 2. Matrix Algebra and Determinants. Basic operations the transpose; diagonal, triangular, and symmetric matrices; invertibility; identities; determinants and properties with some basic proofs.
- 4: Determinants
- Designed for week 3. Determinants are special numbers associated with square matrices. They play an important role in linear algebra because they help answer questions about systems of equations, matrix invertibility, geometry, and transformations. If a matrix is viewed as a machine that transforms vectors, the determinant tells us how that transformation changes area, volume, or higher-dimensional space.
- 5: Rⁿ
- Designed for week 4. Vector algebra in \(\mathbb{R}^n \). This is where we first look into Vector Spaces but a specific Vector Space \{\mathbb{R}^n\). This section tackles vector operations, geometry of vectors, dot products, lines and planes, coordinates, orthogonality of two vectors.
- 5.1: Vectors in Rⁿ
- 5.2: Vector Algebra
- 5.3: Geometric Meaning of Vector Addition
- 5.4: Length of a Vector
- 5.5: Geometric Meaning of Scalar Multiplication
- 5.6: Parametric Lines
- 5.7: The Dot Product
- 5.8: Planes in Rⁿ
- 5.9: The Cross Product
- 5.10: Spanning, Linear Independence and Basis in Rⁿ
- 5.11: Orthogonality
- 5.12: Applications
- 5.E: Exercises
- 6: Vector Spaces
- Designed for week 5. Vector Spaces. Vector Spaces is a broad and abstract concept. In the previous section we saw vector algebra in \(\mathbb{R}^n\. This sections tackles Subspaces; Spans; Independence; Bases; Dimension Theory; change of basis, orthogonal and orthonormal bases.
- 7: Linear Transformations
- Designed for week 6. Linear transformations; matrices of general linear transformations; kernel and range; inverse linear transformations.
- 7.1: Linear Transformations
- 7.2: The Matrix of a Linear Transformation I
- 7.3: Properties of Linear Transformations
- 7.4: Special Linear Transformations in R²
- 7.5: One-to-One and Onto Transformations
- 7.6: Isomorphisms
- 7.7: The Kernel and Image of A Linear Map
- 7.8: The Matrix of a Linear Transformation II
- 7.9: The General Solution of a Linear System
- 7.E: Exercises
- 8: Spectral Theory
- Designed for week 7. Eigenvalues and Eigenvectors; Eigenspace; Diagonalization; matrix exponential function, orthogonal diagonalization.
- 9: First Order Equations
- Designed for week 8, and Introduction to Differential Equations. First-Order Differential Equations, basic definitions and terminology; existence and uniqueness of solutions; separable, linear, homogenous, and exact equations.
- 10: Numerical Methods
- Designed for week 9. Solving first-order differential equations using numerical approximation and the Runge-Kutta Method.
- 11: Applications of First Order Equations
- Designed for week 10. Applications of first Order Equations including slope fields, population models, circuits and orthogonal trajectories.
- 12: Linear Second Order Equations
- Designed for week 11. Higher-Order Linear Differential Equations (namely second order). Covers General theory; Wronskian fundamental solutions; homogenous equations with constant coefficients; non homogeneous equations; method of undetermined coefficients.
- 13: Applications of Linear Second Order Equations
- Designed for week 12. applications of linear second order equations with oscillations of mechanical systems; circuits.
- 14: Linear Higher Order Differential Equations
- Designed for week 13. Higher-order Differential equations involve unknown functions and its derivatives up to order n (the highest derivative determines the order of the equation). This section focuses on Linear DE however they can be nonlinear as well. Additionally, we will discuss homogeneous and non homogeneous. They can be used to model complex physical systems such as vibrations, circuits, beam deflection, and fluid motion.
- 15: Laplace Transforms
- Designed for week 14-15. Laplace transforms, definition, properties, use in solving IVP problems. Basics is to transform a difficult problem into an easier one, solve the latter, and then use its solution to obtain a solution of the original problem. The method discussed here transforms an initial value problem for a constant coefficient equation into an algebraic equation whose solution can then be used to solve the initial value problem.
- 15.1: Introduction to the Laplace Transform
- 15.2: The Inverse Laplace Transform
- 15.3: Solution of Initial Value Problems
- 15.4: The Unit Step Function
- 15.5: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
- 15.6: Convolution
- 15.7: Constant Coefficient Equations with Impulses
- 15.8: A Brief Table of Laplace Transforms