Franklin

Linear algebra / Elizabeth S. Meckes, Case Western Reserve University, Cleveland, OH, USA, Mark W. Meckes, Case Western Reserve University, Cleveland, OH, USA.

Author/Creator:
Meckes, Elizabeth S. author.
Publication:
Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018.
Series:
Cambridge mathematical textbooks
Cambridge mathematical textbooks
Format/Description:
Book
xvi, 427 pages : illustrations (some color) ; 26 cm.
Subjects:
Algebras, Linear -- Textbooks.
Algebras, Linear.
Form/Genre:
Textbooks.
Summary:
Linear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course, which will be useful for classes with a mix of mathematics, physics, engineering, and computer science students. Major topics include singular value decomposition, the spectral theorem, linear systems of equations, vector spaces, linear maps, matrices, eigenvalues and eigenvectors, linear independence, bases, coordinates, dimension, matrix factorizations, inner products, norms, and determinants. -- Provided by publisher.
Contents:
Machine generated contents note: 1.Linear Systems and Vector Spaces
1.1.Linear Systems of Equations
Bread, Beer, and Barley
Linear Systems and Solutions
1.2.Gaussian Elimination
The Augmented Matrix of a Linear System
Row Operations
Does it Always Work?
Pivots and Existence and Uniqueness of Solutions
1.3.Vectors and the Geometry of Linear Systems
Vectors and Linear Combinations
The Vector Form of a Linear System
The Geometry of Linear Combinations
The Geometry of Solutions
1.4.Fields
General Fields
Arithmetic in Fields
Linear Systems over a Field
1.5.Vector Spaces
General Vector Spaces
Examples of Vector Spaces
Arithmetic in Vector Spaces
2.Linear Maps and Matrices
2.1.Linear Maps
Recognizing Sameness
Linear Maps in Geometry
Matrices as Linear Maps
Eigenvalues and Eigenvectors
The Matrix
Vector Form of a Linear System
2.2.More on Linear Maps
Isomorphism
Properties of Linear Maps
Note continued: The Matrix of a Linear Map
Some Linear Maps on Function and Sequence Spaces
2.3.Matrix Multiplication
Definition of Matrix Multiplication
Other Ways of Looking at Matrix Multiplication
The Transpose
Matrix Inverses
2.4.Row Operations and the LU Decomposition
Row Operations and Matrix Multiplication
Inverting Matrices via Row Operations
The LU Decomposition
2.5.Range, Kernel, and Eigenspaces
Range
Kernel
Eigenspaces
Solution Spaces
2.6.Error-correcting Linear Codes
Linear Codes
Error-detecting Codes
Error-correcting Codes
The Hamming Code
3.Linear Independence, Bases, and Coordinates
3.1.Linear (In)dependence
Redundancy
Linear Independence
The Linear Dependence Lemma
Linear Independence of Eigenvectors
3.2.Bases
Bases of Vector Spaces
Properties of Bases
Bases and Linear Maps
3.3.Dimension
The Dimension of a Vector Space
Dimension, Bases, and Subspaces
Note continued: 3.4.Rank and Nullity
The Rank and Nullity of Maps and Matrices
The Rank
Nullity Theorem
Consequences of the Rank
Nullity Theorem
Linear Constraints
3.5.Coordinates
Coordinate Representations of Vectors
Matrix Representations of Linear Maps
Eigenvectors and Diagonalizability
Matrix Multiplication and Coordinates
3.6.Change of Basis
Change of Basis Matrices
Similarity and Diagonalizability
Invariants
3.7.Triangularization
Eigenvalues of Upper Triangular Matrices
Triangularization
4.Inner Products
4.1.Inner Products
The Dot Product in Rn
Inner Product Spaces
Orthogonality
More Examples of Inner Product Spaces
4.2.Orthonormal Bases
Orthonormality
Coordinates in Orthonormal Bases
The Gram-Schmidt Process
4.3.Orthogonal Projections and Optimization
Orthogonal Complements and Direct Sums
Orthogonal Projections
Linear Least Squares
Approximation of Functions
Note continued: 4.4.Normed Spaces
General Norms
The Operator Norm
4.5.Isometries
Preserving Lengths and Angles
Orthogonal and Unitary Matrices
The QR Decomposition
5.Singular Value Decomposition and the Spectral Theorem
5.1.Singular Value Decomposition of Linear Maps
Singular Value Decomposition
Uniqueness of Singular Values
5.2.Singular Value Decomposition of Matrices
Matrix Version of SVD
SVD and Geometry
Low-rank Approximation
5.3.Adjoint Maps
The Adjoint of a Linear Map
Self-adjoint Maps and Matrices
The Four Subspaces
Computing SVD
5.4.The Spectral Theorems
Eigenvectors of Self-adjoint Maps and Matrices
Normal Maps and Matrices
Schur Decomposition
6.Determinants
6.1.Determinants
Multilinear Functions
The Determinant
Existence and Uniqueness of the Determinant
6.2.Computing Determinants
Basic Properties
Determinants and Row Operations
Permutations
Note continued: 6.3.Characteristic Polynomials
The Characteristic Polynomial of a Matrix
Multiplicities of Eigenvalues
The Cayley-Hamilton Theorem
6.4.Applications of Determinants
Volume
Cramer's Rule
Cofactors and Inverses
Appendix
A.1.Sets and Functions
Basic Definitions
Composition and Invertibility
A.2.Complex Numbers
A.3.Proofs
Logical Connectives
Quantifiers
Contrapositives, Counterexamples, and Proof by Contradiction
Proof by Induction.
Notes:
Includes index.
Contributor:
Meckes, Mark W., author.
ISBN:
9781107177901
1107177901
OCLC:
1012749835
Publisher Number:
40028302061
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