# Linear Algebra

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Linear Algebra
An Introduction to Mathematical Discourse

The book was designed specifically for students who had not previously been exposed to mathematics as mathematicians view it. That is, as a subject whose goal is to rigorously prove theorems starting from clear consistent definitions. This book attempts to build students up from a background where mathematics is simply a tool that provides useful calculations to the point where the students have a grasp of the clear and precise nature of mathematics. A more detailed discussion of the prerequisites and goal of this book is given in the introduction.

This book is part of a series on Algebra:

### Linear Systems

1. Solving Linear Systems (Jul 13, 2009)
1. Gauss' Method  (Jul 13, 2009)
2. Describing the Solution Set  (Jul 13, 2009)
3. General = Particular + Homogeneous  (Jul 13, 2009)
4. Comparing Set Descriptions  (Jul 13, 2009)
5. Automation  (Jul 13, 2009)
2. Linear Geometry of n-Space  (Jul 13, 2009)
1. Vectors in Space  (Jul 13, 2009)
2. Length and Angle Measures  (Jul 13, 2009)
3. Reduced Echelon Form  (Jul 13, 2009)
1. Gauss-Jordan Reduction  (Jul 13, 2009)
2. Row Equivalence  (Jul 13, 2009)
4. Topic: Computer Algebra Systems  (Jul 13, 2009)
5. Topic: Input-Output Analysis  (Jul 13, 2009)
6. Input-Output Analysis M File  (Mar 24 2008)
7. Topic: Accuracy of Computations  (Jul 13, 2009)
8. Topic: Analyzing Networks  (Jul 13, 2009)
9. Topic: Speed of Gauss' Method  (Mar 24, 2008)

### Vector Spaces  (Apr 17, 2009)

1. Definition of Vector Space (Apr 17, 2009)
1. Definition and Examples (Jun 18, 2009)
2. Subspaces and Spanning sets (Jun 18, 2009)
2. Linear Independence (Apr 17, 2009)
1. Definition and Examples (Apr 17, 2009)
3. Basis and Dimension (Apr 17, 2009)
1. Basis (Jun 18, 2009)
2. Dimension (Apr 17, 2009)
3. Vector Spaces and Linear Systems (Apr 17, 2009)
4. Combining Subspaces (Apr 17, 2009)
4. Topic: Fields (Apr 17, 2009)
5. Topic: Crystals (Apr 17, 2009)
6. Topic: Voting Paradoxes (Apr 17, 2009)
7. Topic: Dimensional Analysis (Apr 17, 2009)

### Maps Between Spaces

1. Isomorphisms (Jun 21, 2009)
1. Definition and Examples (July 19, 2009)
2. Dimension Characterizes Isomorphism (Jun 21, 2009)
2. Homomorphisms (Jun 21, 2009)
1. Definition of Homomorphism (Jun 21, 2009)
2. Rangespace and Nullspace (Jun 21, 2009)
3. Computing Linear Maps (Jun 21, 2009)
1. Representing Linear Maps with Matrices (Jun 21, 2009)
2. Any Matrix Represents a Linear Map (Jun 21, 2009)
4. Matrix Operations (Jun 21, 2009)
1. Sums and Scalar Products (Jun 21, 2009)
2. Matrix Multiplication (Jun 21, 2009)
3. Mechanics of Matrix Multiplication (Jun 21, 2009)
4. Inverses (Jun 21, 2009)
5. Change of Basis (Jun 21, 2009)
1. Changing Representations of Vectors (Jun 21, 2009)
2. Changing Map Representations (Jun 21, 2009)
6. Projection (Jun 21, 2009)
1. Orthogonal Projection Onto a Line (Jun 21, 2009)
2. Gram-Schmidt Orthogonalization (Jun 21, 2009)
3. Projection Onto a Subspace (Jun 21, 2009)
7. Topic: Line of Best Fit (Jun 21, 2009)
8. Topic: Geometry of Linear Maps (Jun 21, 2009)
9. Topic: Markov Chains (Jun 21, 2009)
10. Topic: Orthonormal Matrices (Jun 21, 2009)

### Determinants (Jun 21, 2009)

1. Definition (Jun 21, 2009)
1. Exploration (Jun 21, 2009)
2. Properties of Determinants (Jun 21, 2009)
3. The Permutation Expansion (Jun 21, 2009)
4. Determinants Exist (Jun 21, 2009)
2. Geometry of Determinants (Jun 21, 2009)
1. Determinants as Size Functions (Jun 21, 2009)
3. Other Formulas for Determinants (Jun 21, 2009)
1. Laplace's Expansion (Jun 21, 2009)
4. Topic: Cramer's Rule (Jun 21, 2009)
5. Topic: Speed of Calculating Determinants (Jun 21, 2009)
6. Topic: Projective Geometry (Jun 21, 2009)

### Similarity (Jun 24, 2009)

1. Complex Vector Spaces (Jun 24, 2009)
1. Factoring and Complex Numbers: A Review (Jun 24, 2009)
2. Complex Representations (Jun 24, 2009)
2. Similarity
1. Definition and Examples (Jun 24, 2009)
2. Diagonalizability (Jun 24, 2009)
3. Eigenvalues and Eigenvectors (Jun 24, 2009)
3. Nilpotence (Jun 24, 2009)
1. Self-Composition (Jun 24, 2009)
2. Strings (Jun 24, 2009)
4. Jordan Form (Jun 24, 2009)
1. Polynomials of Maps and Matrices (Jun 24, 2009)
2. Jordan Canonical Form (Jun 24, 2009)
5. Topic: Geometry of Eigenvalues (Jun 24, 2009)
6. Topic: The Method of Powers (Jun 24, 2009)
7. Topic: Stable Populations (Jun 24, 2009)
8. Topic: Linear Recurrences (Jun 24, 2009)