# Linear Algebra

The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Another standard is book’s audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach — this book's presentation emphasizes motivation and naturalness, using many examples as well as extensive and careful exercises
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• Chapter One: Linear Systems
• I Solving Linear Systems
• I.1 Gauss’s Method
• I.2 Describing the Solution Set
• I.3 General = Particular + Homogeneous
• II Linear Geometry
• II.1 Vectors in Space*
• II.2 Length and Angle Measures*
• III Reduced Echelon Form
• III.1 Gauss-Jordan Reduction
• III.2 The Linear Combination Lemma
• Topic: Computer Algebra Systems
• Topic: Accuracy of Computations
• Topic: Analyzing Networks
• Chapter Two: Vector Spaces
• I Deﬁnition of Vector Space
• I.1 Deﬁnition and Examples
• I.2 Subspaces and Spanning Sets
• II Linear Independence
• II.1 Deﬁnition and Examples
• III Basis and Dimension
• III.1 Basis
• III.2 Dimension
• III.3 Vector Spaces and Linear Systems
• III.4 Combining Subspaces*
• Topic: Fields
• Topic: Crystals
• Topic: Dimensional Analysis
• Chapter Three: Maps Between Spaces
• I Isomorphisms
• I.1 Definition and Examples
• I.2 Dimension Characterizes Isomorphism
• II Homomorphisms
• II.1 Deﬁnition
• II.2 Range space and Null space
• III Computing Linear Maps
• III.1 Representing Linear Maps with Matrices
• III.2 Any Matrix Represents a Linear Map*
• IV Matrix Operations
• IV.1 Sums and Scalar Products
• IV.2 Matrix Multiplication
• IV.3 Mechanics of Matrix Multiplication
• IV.4 Inverses
• V Change of Basis
• V.1 Changing Representations of Vectors
• V.2 Changing Map Representations
• VI Projection
• VI.1 Orthogonal Projection Into a Line*
• VI.2 Gram-Schmidt Orthogonalization*
• VI.3 Projection Into a Subspace*
• Topic: Line of Best Fit
• Topic: Geometry of Linear Maps
• Topic: Magic Squares
• Topic: Markov Chains
• Topic: Orthonormal Matrices
• Chapter Four: Determinants
• I Definition
• I.1 Exploration*
• I.2 Properties of Determinants
• I.3 The Permutation Expansion
• I.4 Determinants Exist*
• II Geometry of Determinants
• II.1 Determinants as Size Functions
• III Laplace’s Formula
• III.1 Laplace’s Expansion*
• Topic: Cramer’s Rule
• Topic: Speed of Calculating Determinants
• Topic: Chiò’s Method
• Topic: Projective Geometry
• Chapter Five: Similarity
• I Complex Vector Spaces
• I.1 Polynomial Factoring and Complex Numbers*
• I.2 Complex Representations
• II Similarity
• II.1 Deﬁnition and Examples
• II.2 Diagonalizability
• II.3 Eigenvalues and Eigenvectors
• III Nilpotence
• III.1 Self-Composition*
• III.2 Strings*
• IV Jordan Form
• IV.1 Polynomials of Maps and Matrices*
• IV.2 Jordan Canonical Form*
• Topic: Method of Powers
• Topic: Stable Populations
• Topic: Page Ranking
• Topic: Linear Recurrences
• Appendix
• Statements
• Quantiﬁers
• Techniques of Proof
• Sets, Functions, and Relations
Supplements:
• Student Solutions Manual. Worked out solutions for selected exercises