Sinopsis
Probably the most important problem in mathematics is that of solving a system of linear equations. Well over 75 percent of all mathematical problems encountered in scientific or industrial applications involve solving a linear system at some stage. By using the methods of modern mathematics, it is often possible to take a sophisticated problem and reduce it to a single system of linear equations. Linear systems arise in applications to such areas as business, economics, sociology, ecology, demography, genetics, electronics, engineering, and physics. Therefore, it seems appropriate to begin this book with a section on linear systems.
Content
- Matrices and Systems of Equations
- Systems of Linear Equations
- Row Echelon Form
- Matrix Arithmetic
- Matrix Algebra
- Elementary Matrices
- Partitioned Matrices
- Determinants
- The Determinant of a Matrix
- Properties of Determinants
- Additional Topics and Applications
- Vector Spaces
- Definition and Examples
- Subspaces
- Linear Independence
- Basis and Dimension
- Change of Basis
- Row Space and Column Space
- Linear Transformations
- Definition and Examples
- Matrix Representations of Linear Transformations
- Similarity
- Orthogonality
- The Scalar Product in Rn
- Orthogonal Subspaces
- Least Squares Problems
- Inner Product Spaces
- Orthonormal Sets
- The Gram–Schmidt Orthogonalization Process
- Orthogonal Polynomials
- Eigenvalues
- Eigenvalues and Eigenvectors
- Diagonalization
- Hermitian Matrices
- The Singular Value Decomposition
- Quadratic Forms
- Positive Definite Matrices
- Nonnegative Matrices
- Numerical Linear Algebra
- Floating-Point Numbers
- Gaussian Elimination
- Pivoting Strategies
- Matrix Norms and Condition Numbers
- Orthogonal Transformations
- The Eigenvalue Problem
- Least Squares Problems
- Iterative Methods
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