Table of Contents

1Systems of Linear Equations

1.1Introduction to Systems of Linear Equations

1.2Augmented Matrix Notation and Elementary Row Operations

1.3Gaussian Elimination and Rank

1.4Iterative Methods for Solving Linear Systems

2Big Ideas about Vectors

2.1Linear Combinations of Vectors

2.2Span

2.3Linear Independence

3Matrices

3.1Addition and Scalar Multiplication of Matrices

3.2Matrix Multiplication

3.3Block Matrix Multiplication

3.4Transpose of a Matrix

3.5Linear Systems as Matrix and Linear Combination Equations

3.6Homogeneous Linear Systems

3.7The Inverse of a Matrix

3.8Elementary Matrices

3.9LU Factorization

4Subspaces of ℝn

4.1ℝn and Subspaces of ℝn

4.2Introduction to Bases

4.3Bases and Dimension

4.4Subspaces of ℝn Associated with Matrices

5Linear Transformations

5.1Matrix Transformations

5.2Geometric Transformations of the Plane

5.3Introduction to Linear Transformations

5.4Standard Matrix of a Linear Transformation from ℝn to ℝm

5.5Composition and Inverses of Linear Transformations

5.6Image and Kernel of a Linear Transformation

6The Determinant

6.1Finding the Determinant

6.2Determinants, Areas, and Volumes

6.3Elementary Row Operations and the Determinant

6.4Properties of the Determinant

6.5Tedious Proofs Concerning Determinants

6.6Determinants and Inverses of Nonsingular Matrices

7Eigenvalues and Eigenvectors

7.1Describing Eigenvalues and Eigenvectors

7.2The Characteristic Equation

7.3Similar Matrices and Their Properties

7.4Diagonalizable Matrices and Multiplicity

7.5Gershgorin’s Theorem

7.6The Power Method and the Dominant Eigenvalue

8Orthogonality

8.1Orthogonality and Projections

8.2Least-Squares Approximation

8.3Gram-Schmidt Orthogonalization

8.4Orthogonal Complements and Decompositions

8.5Orthogonal Matrices and Symmetric Matrices

8.6Positive Definite Matrices

8.7QR Factorization

8.8SVD Decomposition

9Vector Spaces

9.1Abstract Vector Spaces

9.2Bases and Dimension of Abstract Vector Spaces

9.3Linear Transformations of Abstract Vector Spaces

9.4Existence of the Inverse of a Linear Transformation

9.5Isomorphic Vector Spaces

9.6Matrices of Linear Transformations with Respect to Arbitrary Bases

9.7Inner Product Spaces

10Applications

10.1Application to Network Flow

10.2Application to Electrical Networks

10.3Application to Chemical Equations

10.4Application to Input-Output Economic Models

10.5Application to Markov Chains

10.6Curve Fitting

10.7Application to Computer Graphics

11Appendix

11.1Triangle Inequality

11.2Complex Numbers

11.3Complex Matrices

11.4INDEX

11.5Index of GeoGebra Interactives

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