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.

2 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 .. 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.

3 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 .. What is Linear Algebra : 3 3 Matrix Example .. What is Linear Algebra : Hint .. Gaussian Elimination: Augmented Matrix Notation.

4 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.

5 Vectors in Space,n-Vectors: The Story of Your Life .. Vector Spaces: Examples of Each Rule .. 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.

6 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 . 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.

7 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 .. Eigenvalues and Eigenvectors II: Eigenspaces.

8 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.

9 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. 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.

10 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 .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.