Linear Algebra

Description

"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 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