Linear Algebra

1. Vectors, Matrices, and Linear Systems.

 

Vectors in Euclidean Spaces.

 

 

The Norm and the Dot Product.

 

 

Matrices and Their Algebra.

 

 

Solving Systems of Linear Equations.

 

 

Inverses of Square Matrices.

 

 

Homogeneous Systems, Subspaces, and Bases.

 

 

Application to Population Distribution (Optional).

 

 

Application to Binary Linear Codes (Optional).

 

2. Dimension, Rank, and Linear Transformations.

 

Independence and Dimension.

 

 

The Rank of a Matrix.

 

 

Linear Transformations of Euclidean Spaces.

 

 

Linear Transformations of the Plane (Optional).

 

 

Lines, Planes, and Other Flats (Optional).

 

3. Vector Spaces.

 

Vector Spaces.

 

 

Basic Concepts of Vector Spaces.

 

 

Coordinatization of Vectors.

 

 

Linear Transformations.

 

 

Inner-Product Spaces (Optional).

 

4. Determinants.

 

Areas, Volumes, and Cross Products.

 

 

The Determinant of a Square Matrix.

 

 

Computation of Determinants and Cramer's Rule.

 

 

Linear Transformations and Determinants (Optional).

 

5. Eigenvalues and Eigenvectors.

 

Eigenvalues and Eigenvectors.

 

 

Diagonalization.

 

 

Two Applications.

 

6. Orthogonality.

 

Projections.

 

 

The Gram-Schmidt Process.

 

 

Orthogonal Matrices.

 

 

The Projection Matrix.

 

 

The Method of Least Squares.

 

7. Change of Basis.

 

Coordinatization and Change of Basis.

 

 

Matrix Representations and Similarity.

 

8. Eigenvalues: Further Applications and Computations.

 

Diagonalization of Quadratic Forms.

 

 

Applications to Geometry.

 

 

Applications to Extrema.

 

 

Computing Eigenvalues and Eigenvectors.

 

9. Complex Scalars.

 

Algebra of Complex Numbers.

 

 

Matrices and Vector Spaces with Complex Scalars.

 

 

Eigenvalues and Diagonalization.

 

 

Jordan Canonical Form.

 

10. Solving Large Linear Systems.

 

Considerations of Time.

 

 

The LU-Factorization.

 

 

Pivoting, Scaling, and Ill-Conditioned Matrices.

 

Appendices.

 

Mathematical Induction.

 

 

Two Deferred Proofs.

 

 

LINTEK Routines.

 

 

MATLAB Procedures and Commands Used in the Exercises.

 

Appendices.