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Many students of linear algebra hit a wall at mid-semester. Having spent the first part of the term doing mostly computational work, they are unprepared for the rigors of conceptual thinking in an abstract setting that is frequently the focus of the second half of the course. Holt's Linear Algebra with Applications blends computational and conceptual topics throughout. Early treatment of conceptual topics in the context of Euclidean space gives students more time, and a familiar setting, in which to absorb them. This organization also makes it possible to treat eigenvalues and eigenvectors earlier than in most texts. Abstract vector spaces are introduced later, once students have developed a solid conceptual foundation. Concepts and topics are frequently accompanied by applications to provide context and motivation. Because many students learn by example, Linear Algebra with Applications provides a large number of representative examples, over and above those used to introduce topics. The text also has over 2500 exercises, covering computational and conceptual topics over a range of difficulty levels.
Table of Contents
1. Linear Equations 1.1 Lines and Linear Equations 1.2 Linear Equations and Matrices 1.3 Numerical Solutions 1.4 Applications of Linear Systems
2. Euclidean Space 2.1 Vectors 2.2 Span 2.3 Linear Independence
3. Matrices 3.1 Linear Transformations 3.2 Matrix Algebra 3.3 Inverses 3.4 LU Factorization 3.5 Markov Chains
4. Subspaces 4.1 Introduction to Subspaces 4.2 Basis and Dimension 4.3 Row and Column Spaces
5. Determinants 5.1 The Determinant Function 5.2 Properties of the Determinant 5.3 Applications of the Determinant
6. Eigenvalues and Eigenvectors 6.1 Eigenvalues and Eigenvectors 6.2 Iterative Methods 6.3 Change of Basis 6.4 Diagonalization 6.5 Complex Eigenvalues 6.6 Systems of Differential Equations
7. Vector Spaces 7.1 Vector Spaces and Subspaces 7.2 Span and Linear Independence 7.3 Basis and Dimension
8. Orthogonality 8.1 Dot Products and Orthogonal Sets 8.2 Projection and the Gram-Schmidt Process 8.3 Diagonalizing Symmetric Matrices 8.4 Singular Value Decomposition 8.5 Least Squares Regression
9. Linear Transformations 9.1 Definition and Properties 9.2 Isomorphisms 9.3 The Matrix of a Linear Transformation 9.4 Similarity
10. Inner Product Spaces 10.1 Inner Products 10.2 The Gram-Schmidt Process Revisited 10.3 Applications of Inner Products 11. Additional Topics and Applications 11.1 Quadratic Forms 11.2 Positive Definite Matrices 11.3 Constrained Optimization 11.4 Complex Vector Spaces 11.5 Hermitian Matrices