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| Vectors, Matrices, and Linear Systems | |
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| Vectors in Euclidean Spaces | |
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| The Norm and the Dot Product | |
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| Matrices and Their Algebra | |
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| Solving Systems of Linear Equations | |
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| Inverses of Square Matrices | |
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| Homogeneous Systems, Subspaces, and Bases | |
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| Application to Population Distribution (Optional) | |
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| Application to Binary Linear Codes (Optional) | |
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| Dimension, Rank, and Linear Transformations | |
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| Independence and Dimension | |
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| The Rank of a Matrix | |
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| Linear Transformations of Euclidean Spaces | |
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| Linear Transformations of the Plane (Optional) | |
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| Lines, Planes, and Other Flats (Optional) | |
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| Vector Spaces | |
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| Vector Spaces | |
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| Basic Concepts of Vector Spaces | |
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| Coordinatization of Vectors | |
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| Linear Transformations | |
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| Inner-Product Spaces (Optional) | |
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| Determinants | |
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| Areas, Volumes, and Cross Products | |
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| The Determinant of a Square Matrix | |
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| Computation of Determinants and Cramer's Rule | |
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| Linear Transformations and Determinants (Optional) | |
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| Eigenvalues and Eigenvectors | |
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| Eigenvalues and Eigenvectors | |
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| Diagonalization | |
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| Two Applications | |
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| Orthogonality | |
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| Projections | |
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| The Gram-Schmidt Process | |
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| Orthogonal Matrices | |
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| The Projection Matrix | |
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| The Method of Least Squares | |
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| Change of Basis | |
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| Coordinatization and Change of Basis | |
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| Matrix Representations and Similarity | |
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| Eigenvalues: Further Applications and Computations | |
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| Diagonalization of Quadratic Forms | |
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| Applications to Geometry | |
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| Applications to Extrema | |
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| Computing Eigenvalues and Eigenvectors | |
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| Complex Scalars | |
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| Algebra of Complex Numbers | |
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| Matrices and Vector Spaces with Complex Scalars | |
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| Eigenvalues and Diagonalization | |
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| Jordan Canonical Form | |
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| Solving Large Linear Systems | |
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| Considerations of Time | |
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| The LU-Factorization | |
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| Pivoting, Scaling, and Ill-Conditioned Matrices | |
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| Appendices | |
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| Mathematical Induction | |
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| Two Deferred Proofs | |
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| LINTEK Routines | |
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| MATLAB Procedures and Commands Used in the Exercises | |
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| Appendices | |