# Schaum's Outline of Linear Algebra, Fifth Edition

by: Seymour Lipschutz, Marc Lars Lipson

**Abstract:**More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you: 612 fully solved problems, 25 problem-solving videos, concise explanations of all course concepts, and support for all major textbooks for linear algebra courses.

Full details

## Table of Contents

**A.**Preface**B.**List of Symbols**1.**Vectors in Rn and Cn, Spatial Vectors**2.**Algebra of Matrices**3.**Systems of Linear Equations**4.**Vector Spaces**5.**Linear Mappings**6.**Linear Mappings and Matrices**7.**Inner Product Spaces, Orthogonality**8.**Determinants**9.**Diagonalization: Eigenvalues and Eigenvectors**10.**Canonical Forms**11.**Linear Functionals and the Dual Space**12.**Bilinear, Quadratic, and Hermitian Forms**13.**Linear Operators on Inner Product Spaces**A.**Multilinear Products**B.**Algebraic Structures**C.**Polynomials over a Field**D.**Odds and Ends

## Tools & Media

## Expanded Table of Contents

**A.**Preface**B.**List of Symbols**1.**Vectors in Rn and Cn, Spatial Vectors**2.**Algebra of Matrices- Introduction
- Matrices
- Matrix Addition and Scalar Multiplication
- Summation Symbol
- Matrix Multiplication
- Transpose of a Matrix
- Square Matrices
- Powers of Matrices, Polynomials in Matrices
- Invertible (Nonsingular) Matrices
- Special Types of Square Matrices
- Complex Matrices
- Block Matrices
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**3.**Systems of Linear Equations- Introduction
- Basic Definitions, Solutions
- Equivalent Systems, Elementary Operations
- Small Square Systems of Linear Equations
- Systems in Triangular and Echelon Forms
- Gaussian Elimination
- Echelon Matrices, Row Canonical Form, Row Equivalence
- Gaussian Elimination, Matrix Formulation
- Matrix Equation of a System of Linear Equations
- Systems of Linear Equations and Linear Combinations of Vectors
- Homogeneous Systems of Linear Equations
- Elementary Matrices
- LU Decomposition
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**4.**Vector Spaces- Introduction
- Vector Spaces
- Examples of Vector Spaces
- Linear Combinations, Spanning Sets
- Subspaces
- Linear Spans, Row Space of a Matrix
- Linear Dependence and Independence
- Basis and Dimension
- Application to Matrices, Rank of a Matrix
- Sums and Direct Sums
- Coordinates
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**5.**Linear Mappings**6.**Linear Mappings and Matrices**7.**Inner Product Spaces, Orthogonality- Introduction
- Inner Product Spaces
- Examples of Inner Product Spaces
- Cauchy–Schwarz Inequality, Applications
- Orthogonality
- Orthogonal Sets and Bases
- Gram–Schmidt Orthogonalization Process
- Orthogonal and Positive Definite Matrices
- Complex Inner Product Spaces
- Normed Vector Spaces (Optional)
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**8.**Determinants- Introduction
- Determinants of Orders 1 and 2
- Determinants of Order 3
- Permutations
- Determinants of Arbitrary Order
- Properties of Determinants
- Minors and Cofactors
- Evaluation of Determinants
- Classical Adjoint
- Applications to Linear Equations, Cramer's Rule
- Submatrices, Minors, Principal Minors
- Block Matrices and Determinants
- Determinants and Volume
- Determinant of a Linear Operator
- Multilinearity and Determinants
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**9.**Diagonalization: Eigenvalues and Eigenvectors- Introduction
- Polynomials of Matrices
- Characteristic Polynomial, Cayley–Hamilton Theorem
- Diagonalization, Eigenvalues and Eigenvectors
- Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices
- Diagonalizing Real Symmetric Matrices and Quadratic Forms
- Minimal Polynomial
- Characteristic and Minimal Polynomials of Block Matrices
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**10.**Canonical Forms**11.**Linear Functionals and the Dual Space**12.**Bilinear, Quadratic, and Hermitian Forms**13.**Linear Operators on Inner Product Spaces- Introduction
- Adjoint Operators
- Analogy Between A(V) and C, Special Linear Operators
- Self-Adjoint Operators
- Orthogonal and Unitary Operators
- Orthogonal and Unitary Matrices
- Change of Orthonormal Basis
- Positive Definite and Positive Operators
- Diagonalization and Canonical Forms in Inner Product Spaces
- Spectral Theorem
- SOLVED PROBLEMS
- SUPPLEMENTARY PROBLEMS
- ANSWERS TO SUPPLEMENTARY PROBLEMS

**A.**Multilinear Products**B.**Algebraic Structures**C.**Polynomials over a Field**D.**Odds and Ends

**Book Details**

**Title: **Schaum's Outline of Linear Algebra, Fifth Edition

**Publisher: **McGraw-Hill: New York, Chicago, San Francisco, Lisbon, London, Madrid, Mexico City, Milan, New Delhi, San Juan, Seoul, Singapore, Sydney, Toronto

**Copyright / Pub. Date: **2013 The McGraw-Hill Companies, Inc.

**ISBN: **9780071794565

**Authors:****Seymour Lipschutz**
is on the faculty of Temple University and formerly taught at the Polytechnic Institute of Brooklyn. He received his PhD in 1960 at Courant Institute of Mathematical Sciences of New York University. He is one of Schaum's most prolific authors. In particular, he has written, among others, Beginning Linear Algebra, Probability, Discrete Mathematics, Set Theory, Finite Mathematics, and General Topology.
**Marc Lars Lipson**
is on the faculty of the University of Virginia and formerly taught at the University of Georgia, he received his PhD in finance in 1994 from the University of Michigan. He is also the coauthor of Discrete Mathematics and Probability with Seymour Lipschutz.

**Description: **
More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you: 612 fully solved problems, 25 problem-solving videos, concise explanations of all course concepts, and support for all major textbooks for linear algebra courses.