Ebook: Linear Functions and Matrix Theory

Courses that study vectors and elementary matrix theory and introduce linear transformations have proliferated greatly in recent years. Most of these courses are taught at the undergraduate level as part of, or adjacent to, the second-year calculus sequence. Although many students will ultimately find the material in these courses more valuable than calculus, they often experience a class that consists mostly of learning to implement a series of computational algorithms. The objective of this text is to bring a different vision to this course, including many of the key elements called for in current mathematics-teaching reform efforts. Three of the main components of this current effort are the following: 1. Mathematical ideas should be introduced in meaningful contexts, with after a clear understanding formal definitions and procedures developed of practical situations has been achieved. 2. Every topic should be treated from different perspectives, including the numerical, geometric, and symbolic viewpoints. 3. The important ideas need to be visited repeatedly throughout the term, with students' understan9ing deepening each time. This text was written with these three objectives in mind. The first two chapters deal with situations requiring linear functions (at times, locally linear functions) or linear ideas in geometry for their understanding. These situations provide the context in which the formal mathematics is developed, and they are returned to with increasing sophistication throughout the text.

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This textbook is a new introduction to linear algebra for students who have completed the first year of calculus. In the spirit of modern instruction, this elementary presentation of the important ideas in linear algebra emphasizes conceptual understanding, developing applied examples, and working with realistic numerical data before introducing formal mathematical definition and operations. This text emphasizes geometric, symbolic, and numeric presentations of the subject. The first two chapters cover linear phenomena in both numeric and geometric settings. The symbolic manipulation of vectors and matrices is then introduced as a tool for the study of specific problems. Many examples, student exercises, and group project ideas are included.

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This textbook is a new introduction to linear algebra for students who have completed the first year of calculus. In the spirit of modern instruction, this elementary presentation of the important ideas in linear algebra emphasizes conceptual understanding, developing applied examples, and working with realistic numerical data before introducing formal mathematical definition and operations. This text emphasizes geometric, symbolic, and numeric presentations of the subject. The first two chapters cover linear phenomena in both numeric and geometric settings. The symbolic manipulation of vectors and matrices is then introduced as a tool for the study of specific problems. Many examples, student exercises, and group project ideas are included.
Content:
Front Matter....Pages i-xi
Linear Functions....Pages 1-39
Linear Geometry....Pages 40-71
Systems of Linear Equations....Pages 72-116
Basic Matrix Algebra....Pages 117-154
Key Concepts of Linear Algebra in R n ....Pages 155-187
More Vector Geometry....Pages 188-214
Eigenvalues and Eigenvectors of Matrices....Pages 215-245
Matrices as Linear Transformations....Pages 246-276
Orthogonality and Least-Squares Problems....Pages 277-302
Back Matter....Pages 303-330

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This textbook is a new introduction to linear algebra for students who have completed the first year of calculus. In the spirit of modern instruction, this elementary presentation of the important ideas in linear algebra emphasizes conceptual understanding, developing applied examples, and working with realistic numerical data before introducing formal mathematical definition and operations. This text emphasizes geometric, symbolic, and numeric presentations of the subject. The first two chapters cover linear phenomena in both numeric and geometric settings. The symbolic manipulation of vectors and matrices is then introduced as a tool for the study of specific problems. Many examples, student exercises, and group project ideas are included.
Content:
Front Matter....Pages i-xi
Linear Functions....Pages 1-39
Linear Geometry....Pages 40-71
Systems of Linear Equations....Pages 72-116
Basic Matrix Algebra....Pages 117-154
Key Concepts of Linear Algebra in R n ....Pages 155-187
More Vector Geometry....Pages 188-214
Eigenvalues and Eigenvectors of Matrices....Pages 215-245
Matrices as Linear Transformations....Pages 246-276
Orthogonality and Least-Squares Problems....Pages 277-302
Back Matter....Pages 303-330
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