Example: air traffic controller

Linear Algebra in Twenty Five Lectures

Linear Algebra in Twenty Five LecturesTom Denton and Andrew WaldronMarch 27, 2012 Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw1 Contents1 What is Linear Algebra ?122 Gaussian Notation for Linear Systems .. Reduced Row Echelon Form .. 213 Elementary Row Operations274 Solution Sets for Systems of Linear Non-Leading Variables .. 355 Vectors in Space, Directions and Magnitudes .. 466 Vector Spaces537 Linear Transformations588 Matrices639 Properties of Block Matrices .. The Algebra of Square Matrices.. 7310 Inverse Three Properties of the Inverse .. Finding Inverses .. Linear Systems and Inverses .. Homogeneous Systems .. Bit Matrices .. UsingLUDecomposition to Solve Linear Systems .. Finding anLUDecomposition.. BlockLDUD ecomposition .. 94212 Elementary Matrices and Permutations .. Elementary Matrices .. 10013 Elementary Matrices and Determinants II10714 Properties of the Determinant of the Inverse.

Introductory Video The notes are designed to be used in conjunction with a set of online homework exercises which help the students read the lecture notes and learn basic linear algebra skills. Interspersed among the lecture notes are links to simple online problems that test whether students are actively reading the notes.

Tags:

  Notes, Introductory

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Linear Algebra in Twenty Five Lectures

1 Linear Algebra in Twenty Five LecturesTom Denton and Andrew WaldronMarch 27, 2012 Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw1 Contents1 What is Linear Algebra ?122 Gaussian Notation for Linear Systems .. Reduced Row Echelon Form .. 213 Elementary Row Operations274 Solution Sets for Systems of Linear Non-Leading Variables .. 355 Vectors in Space, Directions and Magnitudes .. 466 Vector Spaces537 Linear Transformations588 Matrices639 Properties of Block Matrices .. The Algebra of Square Matrices.. 7310 Inverse Three Properties of the Inverse .. Finding Inverses .. Linear Systems and Inverses .. Homogeneous Systems .. Bit Matrices .. UsingLUDecomposition to Solve Linear Systems .. Finding anLUDecomposition.. BlockLDUD ecomposition .. 94212 Elementary Matrices and Permutations .. Elementary Matrices .. 10013 Elementary Matrices and Determinants II10714 Properties of the Determinant of the Inverse.

2 Adjoint of a Matrix .. Application: Volume of a Parallelepiped .. 12215 Subspaces and Spanning Subspaces .. Building Subspaces .. 12616 Linear Independence13117 Basis and Bases inRn.. 14218 Eigenvalues and Matrix of a Linear Transformation .. Invariant Directions .. 15119 Eigenvalues and Eigenvectors Eigenspaces .. 16220 Diagonalization .. Change of Basis .. 16621 Orthonormal Relating Orthonormal Bases .. 17622 Gram-Schmidt and Orthogonal Orthogonal Complements .. 18523 Diagonalizing Symmetric Matrices191324 Kernel, Range, Nullity, Summary .. 20125 Least Squares206A Sample Midterm I Problems and Solutions211B Sample Midterm II Problems and Solutions221C Sample Final Problems and Solutions231D Points Vs. Vectors256E Abstract Dual Spaces .. Groups .. Fields .. Rings .. Algebras .. 261F Sine and Cosine as an Orthonormal Basis262G Movie introductory Video .. What is Linear Algebra : Overview.

3 What is Linear Algebra : 3 3 Matrix Example .. What is Linear Algebra : Hint .. Gaussian Elimination: Augmented Matrix Notation .. Gaussian Elimination: Equivalence of Augmented Matrices .. Gaussian Elimination: Hints for Review Questions 4 and 5 .. Gaussian Elimination: 3 3 Example .. Elementary Row Operations: Example .. Elementary Row Operations: Worked Examples .. Elementary Row Operations: Explanation of Proof for Theo-rem .. Elementary Row Operations: Hint for Review Question 3 .. Solution Sets for Systems of Linear Equations: Planes .. Solution Sets for Systems of Linear Equations: Pictures andExplanation .. Solution Sets for Systems of Linear Equations: Example .. Solution Sets for Systems of Linear Equations: Hint .. Vectors in Space,n-Vectors: Overview .. Vectors in Space,n-Vectors: Review of Parametric Notation . Vectors in Space,n-Vectors: The Story of Your Life .. Vector Spaces: Examples of Each Rule.

4 Vector Spaces: Example of a Vector Space .. Vector Spaces: Hint .. Linear Transformations: A Linear and A Non- Linear Example Linear Transformations: Derivative and Integral of (Real) Poly-nomials of Degree at Most 3 .. Linear Transformations: Linear Transformations Hint .. Matrices: Adjacency Matrix Example .. Matrices: Do Matrices Commute? .. Matrices: Hint for Review Question 4 .. Matrices: Hint for Review Question 5 .. Properties of Matrices: Matrix Exponential Example .. Properties of Matrices: Explanation of the Proof .. Properties of Matrices: A Closer Look at the Trace Function . Properties of Matrices: Matrix Exponent Hint .. Inverse Matrix: A 2 2 Example .. Inverse Matrix: Hints for Problem 3 .. Inverse Matrix: Left and Right Inverses .. : Example: How to Use LU Decomposition : Worked Example .. : BlockLDUE xplanation .. Elementary Matrices and Determinants: Permutations .. Elementary Matrices and Determinants: Some Ideas Explained Elementary Matrices and Determinants: Hints for Problem 4.

5 Elementary Matrices and Determinants II: Elementary Deter-minants .. Elementary Matrices and Determinants II: Determinants andInverses .. Elementary Matrices and Determinants II: Product of Deter-minants .. Properties of the Determinant: Practice taking Determinants Properties of the Determinant: The Adjoint Matrix .. Properties of the Determinant: Hint for Problem 3 .. Subspaces and Spanning Sets: Worked Example .. Subspaces and Spanning Sets: Hint for Problem 2 .. Subspaces and Spanning Sets: Hint .. Linear Independence: Worked Example .. Linear Independence: Proof of Theorem .. Linear Independence: Hint for Problem 1 .. Basis and Dimension: Proof of Theorem .. Basis and Dimension: Worked Example .. Basis and Dimension: Hint for Problem 2 .. Eigenvalues and Eigenvectors: Worked Example .. Eigenvalues and Eigenvectors: 2 2 Example .. Eigenvalues and Eigenvectors: Jordan Cells .. Eigenvalues and Eigenvectors II: Eigenvalues.

6 Eigenvalues and Eigenvectors II: Eigenspaces .. Eigenvalues and Eigenvectors II: Hint .. Diagonalization: Derivative Is Not Diagonalizable .. Diagonalization: Change of Basis Example .. Diagonalization: Diagionalizing Example .. Orthonormal Bases: Sine and Cosine Form All OrthonormalBases forR2.. Orthonormal Bases: Hint for Question 2, Lecture 21 .. Orthonormal Bases: Hint.. Gram-Schmidt and Orthogonal Complements: 4 4 GramSchmidt Example .. Gram-Schmidt and Orthogonal Complements: Overview .. Gram-Schmidt and Orthogonal Complements: QR Decompo-sition Example .. Gram-Schmidt and Orthogonal Complements: Hint for Prob-lem 1 .. Diagonalizing Symmetric Matrices: 3 3 Example .. Diagonalizing Symmetric Matrices: Hints for Problem 1 .. Kernel, Range, Nullity, Rank: Invertibility Conditions .. Kernel, Range, Nullity, Rank: Hint for 1.. Least Squares: Hint for Problem 1 .. Least Squares: Hint for Problem 2 .. 388H Student Contributions3896I Other Resources390J List of Symbols392 Index3937 PrefaceThese Linear Algebra lecture notes are designed to be presented as Twenty five,fifty minute Lectures suitable for sophomores likely to use the material forapplications but still requiring a solid foundation in this fundamental branchof mathematics.

7 The main idea of the course is to emphasize the conceptsof vector spaces and Linear transformations as mathematical structures thatcan be used to model the world around us. Once persuaded of this truth,students learn explicit skills such as Gaussian elimination and diagonalizationin order that vectors and Linear transformations become calculational tools,rather than abstract practical terms, the course aims to produce students who can performcomputations with large Linear systems while at the same time understandthe concepts behind these techniques. Often-times when a problem can be re-duced to one of Linear Algebra it is solved . These notes do not devote muchspace to applications (there are already a plethora of textbooks with titlesinvolving some permutation of the words Linear , Algebra and applica-tions ). Instead, they attempt to explain the fundamental concepts carefullyenough that students will realize for their own selves when the particularapplication they encounter in future studies is ripe for a solution via are relatively few worked examples or illustrations in these notes ,this material is instead covered by a series of Linear Algebra how-to videos.

8 They can be viewed by clicking on the take one icon. The scripts for these movies are found at the end of the notes if students prefer to readthis material in a traditional format and can be easily reached via the scripticon. Watch an introductory video below: introductory VideoThe notes are designed to be used in conjunction with a set of onlinehomework exercises which help the students read the lecture notes and learnbasic Linear Algebra skills. Interspersed among the lecture notes are linksto simple online problems that test whether students are actively readingthe notes . In addition there are two sets of sample midterm problems withsolutions as well as a sample final exam. There are also a set of ten on-line assignments which are usually collected weekly. The first assignment8is designed to ensure familiarity with some basic mathematic notions (sets,functions, logical quantifiers and basic methods of proof). The remainingnine assignments are devoted to the usual matrix and vector gymnasticsexpected from any sophomore Linear Algebra class.

9 These exercises are allavailable is an open source, online homework system which originated atthe University of Rochester. It can efficiently check whether a student hasanswered an explicit, typically computation-based, problem correctly. Theproblem sets chosen to accompany these notes could contribute roughly 20%of a student s grade, and ensure that basic computational skills are students rapidly realize that it is best to print out the Webwork assign-ments and solve them on paper before entering the answers online. Thosewho do not tend to fare poorly on midterm examinations. We have foundthat there tend to be relatively few questions from students in office hoursabout the Webwork assignments. Instead, by assigning 20% of the gradeto written assignments drawn from problems chosen randomly from the re-view exercises at the end of each lecture, the student s focus was primarilyon understanding ideas. They range from simple tests of understanding ofthe material in the Lectures to more difficult problems, all of them requirethinking, rather than blind application of mathematical recipes.

10 Officehour questions reflected this and offered an excellent chance to give studentstips how to present written answers in a way that would convince the persongrading their work that they deserved full credit!Each lecture concludes with references to the comprehensive online text-books of Jim Hefferon and Rob Beezer: the notes are also hyperlinked to Wikipedia where students can rapidlyaccess further details and background material for many of the of Linear Algebra Lectures are available online from at least two sources: The Khan Academy, # Linear Algebra9 MIT OpenCourseWare, Professor Gilbert Strang, are also an array of useful commercially available texts. A non-exhaustive list includes introductory Linear Algebra , An Applied First Course , B. Kolmanand D. Hill, Pearson 2001. Linear Algebra and Its Applications , David C. Lay, Addison Weseley2011. Introduction to Linear Algebra , Gilbert Strang, Wellesley CambridgePress 2009. Linear Algebra Done Right , S.


Related search queries