Table of Contents

1Preliminaries

1.1A Brief Introduction to ℝn

1.2Introduction to Vectors

1.3Length of a Vector

1.4Vector Arithmetic

1.5Standard Unit Vectors in ℝn

1.6Unit Vector in the Direction of a Given Vector

1.7Dot Product and its Properties

1.8Dot Product and Angle

1.9Orthogonal Projections

1.10Cross Product and its Properties

1.11Parametric Equations of Lines

1.12Planes in ℝ3

2Systems of Linear Equations

2.1Introduction to Systems of Linear Equations

2.2Augmented Matrix Notation and Elementary Row Operations

2.3Gaussian Elimination and Rank

2.4Iterative Methods for Solving Linear Systems

3Big Ideas about Vectors

3.1Linear Combinations of Vectors

3.2Span

3.3Linear Independence

4Matrices

4.1Addition and Scalar Multiplication of Matrices

4.2Matrix Multiplication

4.3Block Matrix Multiplication

4.4Transpose of a Matrix

4.5Linear Systems as Matrix and Linear Combination Equations

4.6Homogeneous Linear Systems

4.7The Inverse of a Matrix

4.8Elementary Matrices

4.9LU Factorization

5Subspaces of ℝn

5.1ℝn and Subspaces of ℝn

5.2Introduction to Bases

5.3Bases and Dimension

5.4Subspaces of ℝn Associated with Matrices

6Linear Transformations

6.1Matrix Transformations

6.2Geometric Transformations of the Plane

6.3Introduction to Linear Transformations

6.4Standard Matrix of a Linear Transformation from ℝn to ℝm

6.5Composition and Inverses of Linear Transformations

6.6Image and Kernel of a Linear Transformation

7The Determinant

7.1Finding the Determinant

7.2Determinants, Areas, and Volumes

7.3Elementary Row Operations and the Determinant

7.4Properties of the Determinant

7.5Tedious Proofs Concerning Determinants

7.6Determinants and Inverses of Nonsingular Matrices

8Eigenvalues and Eigenvectors

8.1Describing Eigenvalues and Eigenvectors

8.2The Characteristic Equation

8.3Similar Matrices and Their Properties

8.4Diagonalizable Matrices and Multiplicity

8.5Gershgorin’s Theorem

8.6The Power Method and the Dominant Eigenvalue

9Orthogonality

9.1Orthogonality and Projections

9.2Least-Squares Approximation

9.3Gram-Schmidt Orthogonalization

9.4Orthogonal Complements and Decompositions

9.5Orthogonal Matrices and Symmetric Matrices

9.6Positive Definite Matrices

9.7QR Factorization

9.8SVD Decomposition

10Vector Spaces

10.1Abstract Vector Spaces

10.2Bases and Dimension of Abstract Vector Spaces

10.3Linear Transformations of Abstract Vector Spaces

10.4Existence of the Inverse of a Linear Transformation

10.5Isomorphic Vector Spaces

10.6Matrices of Linear Transformations with Respect to Arbitrary Bases

11Applications

11.1Application to Network Flow

11.2Application to Electrical Networks

11.3Application to Chemical Equations

11.4Application to Input-Output Economic Models

11.5Application to Markov Chains

11.6Curve Fitting

11.7Application to Computer Graphics

12Appendix

12.1Triangle Inequality

12.2Complex Numbers

12.3Complex Matrices

12.4Inner Product Spaces

12.5INDEX

12.6Index of GeoGebra Interactives

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