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Chapter 1 Linear Systems and Matrices
1.1 Introduction to Linear Systems and Matrices
1.1.1 Linear equations and linear systems
1.1.2 Matrices
1.1.3 Elementary row operations
1.2 Gauss-Jordan Elimination
1.2.1 Reduced row-echelon form
1.2.2 Gauss-Jordan elimination
1.2.3 Homogeneous linear systems
1.3 Matrix Operations
1.3.1 Operations on matrices
1.3.2 Partition of matrices
1.3.3 Matrix product by columns and by rows
1.3.4 Matrix product of partitioned matrices
1.3.5 Matrix form of a linear system
1.3.6 Transpose and trace of a matrix
1.4 Rules of Matrix Operations and Inverses
1.4.1 Basic properties of matrix operations
1.4.2 Identity matrix and zero matrix
1.4.3 Inverse of a matrix
1.4.4 Powers of a matrix
1.5 Elementary Matrices and a Method for Finding A-1
1.5.1 Elementary matrices and their properties
1.5.2 Main theorem of invertibility
1.5.3 A method for finding A-1
1.6 Further Results on Systems and Invertibility
1.6.1 A basic theorem
1.6.2 Properties of invertible matrices
1.7 Some Special Matrices
1.7.1 Diagonal and triangular matrices
1.7.2 Symmetric matrix
Exercises
Chapter 2 Determinants
2.1 Determinant Function
2.1.1 Permutation, inversion, and elementary product
2.1.2 Definition of determinant function
2.2 Evaluation of Determinants
2.2.1 Elementary theorems
2.2.2 A method for evaluating determinants
2.3 Properties of Determinants
2.3.1 Basic properties
2.3.2 Determinant of a matrix product
2.3.3 Summary
2.4 Cofactor Expansions and Cramer’s Rule
2.4.1 Cofactors
2.4.2 Cofactor expansions
2.4.3 Adjoint of a matrix
2.4.4 Cramer’s rule
Exercises
Chapter 3 Euclidean Vector Spaces
3.1 Euclidean n-Space
3.1.1 n-vector space
3.1.2 Euclidean n-space
3.1.3 Norm, distance, angle, and orthogonality
3.1.4 Some remarks
3.2 Linear Transformations from Rn to Rm
3.2.1 Linear transformations from Rn to Rm
3.2.2 Some important linear transformations
3.2.3 Compositions of linear transformations
3.3 Properties of Transformations
3.3.1 Linearity conditions
3.3.2 Example
3.3.3 One-to-one transformations
3.3.4 Summary
Exercises
Chapter 4 General Vector Spaces
4.1 Real Vector Spaces
4.1.1 Vector space axioms
4.1.2 Some properties
4.2 Subspaces
4.2.1 Definition of subspace
4.2.2 Linear combinations
4.3 Linear Independence
4.3.1 Linear independence and linear dependence
4.3.2 Some theorems
4.4 Basis and Dimension
4.4.1 Basis for vector space
4.4.2 Coordinates
4.4.3 Dimension
4.4.4 Some fundamental theorems
4.4.5 Dimension theorem for subspaces
4.5 Row Space, Column Space, and Nullspace
4.5.1 Definition of row space, column space, and nullspace
4.5.2 Relation between solutions of Ax = 0 and Ax=b
4.5.3 Bases for three spaces
4.5.4 A procedure for finding a basis for span(S)
4.6 Rank and Nullity
4.6.1 Rank and nullity
4.6.2 Rank for matrix operations
4.6.3 Consistency theorems
4.6.4 Summary
Exercises
Chapter 5 Inner Product Spaces
5.1 Inner Products
5.1.1 General inner products
5.1.2 Examples
5.2 Angle and Orthogonality
5.2.1 Angle between two vectors and orthogonality
5.2.2 Properties of length, distance, and orthogonality
5.2.3 Complement
5.3 Orthogonal Bases and Gram-Schmidt Process
5.3.1 Orthogonal and orthonormal bases
5.3.2 Projection theorem
5.3.3 Gram-Schmidt process
5.3.4 QR-decomposition
5.4 Best Approximation and Least Squares
5.4.1 Orthogonal projections viewed as approximations
5.4.2 Least squares solutions of linear systems
5.4.3 Uniqueness of least squares solut