Preface.

Conventions and Notations.

**1. An Introduction to Mathematica.**

1.1 The Very Basics.

1.2 Basic Arithmetic.

1.3 Lists and Matrices.

1.4 Expressions Versus Functions.

1.5 Plotting and Animations.

1.6 Solving Systems of Equations.

1.7 Basic Programming.

**2. Linear Systems of Equations and Matrices.**

2.1 Linear Systems of Equations.

2.2 Augmented Matrix of a Linear System and Row Operations.

2.3 Some Matrix Arithmetic.

**3. Gauss-Jordan Elimination and Reduced Row Echelon Form**.

3.1 Gauss-Jordan Elimination and *rref*.

3.2 Elementary Matrices.

3.3 Sensitivity of Solutions to Error in the Linear System.

**4. Applications of Linear Systems and Matrices.**

4.1 Applications of Linear Systems to Geometry.

4.2 Applications of Linear Systems to Curve Fitting.

4.3 Applications of Linear Systems to Economics.

4.4 Applications of Matrix Multiplication to Geometry.

4.5 An Application of Matrix Multiplication to Economics.

**5. Determinants, Inverses, and Cramer’ Rule.**

5.1 Determinants and Inverses from the Adjoint Formula.

5.2 Determinants by Expanding Along Any Row or Column.

5.3 Determinants Found by Triangularizing Matrices.

5.4 LU Factorization.

5.5 Inverses from *rref*.

5.6 Cramer’s Rule.

**6. Basic Linear Algebra Topics.**

6.1 Vectors.

6.2 Dot Product.

6.3 Cross Product.

6.4 A Vector Projection.

**7. A Few Advanced Linear Algebra Topics.**

7.1 Rotations in Space.

7.2 “Rolling” a Circle Along a Curve.

7.3 The TNB Frame.

**8. Independence, Basis, and Dimension for Subspaces of R**_{n}.

8.1 Subspaces of R_{n}.

8.2 Independent and Dependent Sets of Vectors in R_{n}.

8.3 Basis and Dimension for Subspaces of R_{n}.

8.4 Vector Projection onto a subspace of R_{n}.

8.5 The Gram-Schmidt Orthonormalization Process.

**9. Linear Maps from R**_{n} to R_{m}.

9.1 Basics About Linear Maps.

9.2 The Kernel and Image Subspaces of a Linear Map.

9.3 Composites of Two Linear Maps and Inverses.

9.4 Change of Bases for the Matrix Representation of a Linear Map.

**10. The Geometry of Linear and Affine Maps.**

10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions.

10.2 The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in R_{2}.

10.3 The Effect of Linear Maps on Volume, Area, and Arclength in R_{3}.

10.4 Rotations, Reflections, and Rescalings in Three Dimensions.

10.5 Affine Maps.

**11. Least-Squares Fits and Pseudoinverses.**

11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear System.

11.2 Fits and Pseudoinverses.

11.3 Least-Squares Fits and Pseudoinverses.

**12. Eigenvalues and Eigenvectors.**

12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?

12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix.

12.3 Applications of the Diagonalizability of Square Matrices.

12.4 Solving a Square First-Order Linear System if Differential Equations.

12.5 Basic Facts About Eigenvalues, Eigenvectors, and Diagonalizability.

12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors.

12.7 A Mathematica EigenFunction.

Suggested Reading.

Indices.

Keyword Index.

Index of Mathematica Commands.