Linear algebra done right / Sheldon Axler.

Format:

Book

Author/Creator:

Axler, Sheldon Jay, author.

Series:

Undergraduate texts in mathematics 0172-6056

Undergraduate texts in mathematics, 0172-6056

Language:

English

Subjects (All):

Algebras, Linear.

Algebras, Linear--Study and teaching.

Physical Description:

xvii, 340 pages : color illustrations ; 25 cm.

Edition:

Third edition.

Place of Publication:

Cham : Springer, [2015]

Summary:

"This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra." -- Back cover.

Contents:

Machine generated contents note: 1.Vector Spaces

1.A.Rn and Cn

Complex Numbers

Lists

Fn

Digression on Fields

Exercises 1.A

1.B.Definition of Vector Space

Exercises 1.B

1.C.Subspaces

Sums of Subspaces

Direct Sums

Exercises 1.C

2.Finite-Dimensional Vector Spaces

2.A.Span and Linear Independence

Linear Combinations and Span

Linear Independence

Exercises 2.A

2.B.Bases

Exercises 2.B

2.C.Dimension

Exercises 2.C

3.Linear Maps

3.A.The Vector Space of Linear Maps

Definition and Examples of Linear Maps

Algebraic Operations on L(V, W)

Exercises 3.A

3.B.Null Spaces and Ranges

Null Space and Injectivity

Range and Surjectivity

Fundamental Theorem of Linear Maps

Exercises 3.B

3.C.Matrices

Representing a Linear Map by a Matrix

Addition and Scalar Multiplication of Matrices

Matrix Multiplication

Exercises 3.C

3.D.Invertibility and Isomorphic Vector Spaces

Invertible Linear Maps

Note continued: Isomorphic Vector Spaces

Linear Maps Thought of as Matrix Multiplication

Operators

Exercises 3.D

3.E.Products and Quotients of Vector Spaces

Products of Vector Spaces

Products and Direct Sums

Quotients of Vector Spaces

Exercises 3.E

3.F.Duality

The Dual Space and the Dual Map

The Null Space and Range of the Dual of a Linear Map

The Matrix of the Dual of a Linear Map

The Rank of a Matrix

Exercises 3.F

4.Polynomials

Complex Conjugate and Absolute Value

Uniqueness of Coefficients for Polynomials

The Division Algorithm for Polynomials

Zeros of Polynomials

Factorization of Polynomials over C

Factorization of Polynomials over R

Exercises 4

5.Eigenvalues, Eigenvectors, and Invariant Subspaces

5.A.Invariant Subspaces

Eigenvalues and Eigenvectors

Restriction and Quotient Operators

Exercises 5.A

5.B.Eigenvectors and Upper-Triangular Matrices

Polynomials Applied to Operators

Note continued: Existence of Eigenvalues

Upper-Triangular Matrices

Exercises 5.B

5.C.Eigenspaces and Diagonal Matrices

Exercises 5.C

6.Inner Product Spaces

6.A.Inner Products and Norms

Inner Products

Norms

Exercises 6.A

6.B.Orthonormal Bases

Linear Functionals on Inner Product Spaces

Exercises 6.B

6.C.Orthogonal Complements and Minimization Problems

Orthogonal Complements

Minimization Problems

Exercises 6.C

7.Operators on Inner Product Spaces

7.A.Self-Adjoint and Normal Operators

Adjoints

Self-Adjoint Operators

Normal Operators

Exercises 7.A

7.B.The Spectral Theorem

The Complex Spectral Theorem

The Real Spectral Theorem

Exercises 7.B

7.C.Positive Operators and Isometries

Positive Operators

Isometries

Exercises 7.C

7.D.Polar Decomposition and Singular Value Decomposition

Polar Decomposition

Singular Value Decomposition

Exercises 7.D

Note continued: 8.Operators on Complex Vector Spaces

8.A.Generalized Eigenvectors and Nilpotent Operators

Null Spaces of Powers of an Operator

Generalized Eigenvectors

Nilpotent Operators

Exercises 8.A

8.B.Decomposition of an Operator

Description of Operators on Complex Vector Spaces

Multiplicity of an Eigenvalue

Block Diagonal Matrices

Square Roots

Exercises 8.B

8.C.Characteristic and Minimal Polynomials

The Cayley

Hamilton Theorem

The Minimal Polynomial

Exercises 8.C

8.D.Jordan Form

Exercises 8.D

9.Operators on Real Vector Spaces

9.A.Complexification

Complexification of a Vector Space

Complexification of an Operator

The Minimal Polynomial of the Complexification

Eigenvalues of the Complexification

Characteristic Polynomial of the Complexification

Exercises 9.A

9.B.Operators on Real Inner Product Spaces

Normal Operators on Real Inner Product Spaces

Note continued: Isometries on Real Inner Product Spaces

Exercises 9.B

10.Trace and Determinant

10.A.Trace

Change of Basis

Trace: A Connection Between Operators and Matrices

Exercises 10.A

10.B.Determinant

Determinant of an Operator

Determinant of a Matrix

The Sign of the Determinant

Volume

Exercises 10.B.

Notes:

Includes indexes.

Other Format:

Online version: Axler, Sheldon Jay. Linear algebra done right.

ISBN:

9783319110790

3319110799

3319110802

9783319110806

9783319307657

3319307657

OCLC:

899978064