# 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

License:

**Creative Commons Attribution Sharealike**. This license is considered to be some to be the most open license. It allows reuse, remixing, and distribution (including commercial), but requires any remixes use the same license as the original. This limits where the content can be remixed into, but on the other hand ensures that no-one can remix the content then put the remix under a more restrictive license.Formats:

**PDF**. A Portable Document Format (PDF) file is can be opened using the free Acrobat Reader. It is not an editable format.**TeX**. A TeX file use the TeX or LaTeX typesetting engine. TeX software is available free for most platforms. It is an editable format

Openness Rating (0-4): 3

- 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: Voting Paradoxes
- 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