"This "book" grew out of a series of twenty five lecture notes for a sophomore linear algebra class taught at the [University] of California, Davis. The [audience] was primarily engineering students and students of pure sciences, some of whom may go on to major in mathematics. It was motivated by the lack of a book that taught students basic structures of linear algebra without [overdoing] mathematical rigor or becoming a mindless exercise in crunching recipes at the cost of fundamental understanding. In particular we wanted a book that was suitable for all students, not just math majors, that [focused] on concepts and developing the ability to think in terms of abstract structures in order to address the dizzying array of seemingly disparate applications that can all actually be addressed with linear algebra [methods]."--Back cover
Author(s): David Cherney; Tom Denton; Andrew Kenneth Waldron
Publisher: CreateSpace Independent Publishing Platform
Year: 2016
Language: English
Pages: 398
Tags: Linear Algebra; Vector Spaces; Matrices; Determinants; Linear Independence; Eigenvalues; Eigenvectors; Diagonalization
What is Linear Algebra?
Organizing Information
What are Vectors?
What are Linear Functions?
So, What is a Matrix?
Matrix Multiplication is Composition of Functions
The Matrix Detour
Review Problems
Systems of Linear Equations
Gaussian Elimination
Augmented Matrix Notation
Equivalence and the Act of Solving
Reduced Row Echelon Form
Solution Sets and RREF
Review Problems
Elementary Row Operations
EROs and Matrices
Recording EROs in ( M | I )
The Three Elementary Matrices
LU, LDU, and PLDU Factorizations
Review Problems
Solution Sets for Systems of Linear Equations
The Geometry of Solution Sets: Hyperplanes
Particular Solution + Homogeneous Solutions
Solutions and Linearity
Review Problems
The Simplex Method
Pablo's Problem
Graphical Solutions
Dantzig's Algorithm
Pablo Meets Dantzig
Review Problems
Vectors in Space, n-Vectors
Addition and Scalar Multiplication in Rn
Hyperplanes
Directions and Magnitudes
Vectors, Lists and Functions: RS
Review Problems
Vector Spaces
Examples of Vector Spaces
Non-Examples
Other Fields
Review Problems
Linear Transformations
The Consequence of Linearity
Linear Functions on Hyperplanes
Linear Differential Operators
Bases (Take 1)
Review Problems
Matrices
Linear Transformations and Matrices
Basis Notation
From Linear Operators to Matrices
Review Problems
Properties of Matrices
Associativity and Non-Commutativity
Block Matrices
The Algebra of Square Matrices
Trace
Review Problems
Inverse Matrix
Three Properties of the Inverse
Finding Inverses (Redux)
Linear Systems and Inverses
Homogeneous Systems
Bit Matrices
Review Problems
LU Redux
Using LU Decomposition to Solve Linear Systems
Finding an LU Decomposition.
Block LDU Decomposition
Review Problems
Determinants
The Determinant Formula
Simple Examples
Permutations
Elementary Matrices and Determinants
Row Swap
Row Multiplication
Row Addition
Determinant of Products
Review Problems
Properties of the Determinant
Determinant of the Inverse
Adjoint of a Matrix
Application: Volume of a Parallelepiped
Review Problems
Subspaces and Spanning Sets
Subspaces
Building Subspaces
Review Problems
Linear Independence
Showing Linear Dependence
Showing Linear Independence
From Dependent Independent
Review Problems
Basis and Dimension
Bases in Rn.
Matrix of a Linear Transformation (Redux)
Review Problems
Eigenvalues and Eigenvectors
Invariant Directions
The Eigenvalue–Eigenvector Equation
Eigenspaces
Review Problems
Diagonalization
Diagonalizability
Change of Basis
Changing to a Basis of Eigenvectors
Review Problems
Orthonormal Bases and Complements
Properties of the Standard Basis
Orthogonal and Orthonormal Bases
Orthonormal Bases and Dot Products
Relating Orthonormal Bases
Gram-Schmidt & Orthogonal Complements
The Gram-Schmidt Procedure
QR Decomposition
Orthogonal Complements
Review Problems
Diagonalizing Symmetric Matrices
Review Problems
Kernel, Range, Nullity, Rank
Range
Image
One-to-one and Onto
Kernel
Summary
Review Problems
Least squares and Singular Values
Projection Matrices
Singular Value Decomposition
Review Problems
List of Symbols
Fields
Online Resources
Sample First Midterm
Sample Second Midterm
Sample Final Exam
Movie Scripts
What is Linear Algebra?
Systems of Linear Equations
Vectors in Space n-Vectors
Vector Spaces
Linear Transformations
Matrices
Determinants
Subspaces and Spanning Sets
Linear Independence
Basis and Dimension
Eigenvalues and Eigenvectors
Diagonalization
Orthonormal Bases and Complements
Diagonalizing Symmetric Matrices
Kernel, Range, Nullity, Rank
Least Squares and Singular Values
Index
