# Alignment: Schaum’s Outlines • Linear Algebra • Sixth Edition

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided below. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

• Chapter 1 Vectors in 𝐑^𝑛 and 𝐂^𝑛, Spatial Vectors
• 1.1 Introduction
• 1.2 Vectors in 𝐑^𝑛
• 1.3 Vector Addition and Scalar Multiplication
• 1.4 Dot (Inner) Product
• 1.5 Located Vectors, Hyperplanes, Lines, Curves in 𝐑^𝑛
• 1.6 Vectors in 𝐑³ (Spatial Vectors), 𝐢𝐣𝐤 Notation
• 1.7 Complex Numbers
• 1.8 Vectors in 𝐂^𝑛
• Chapter 2 Algebra of Matrices
• 2.1 Introduction
• 2.2 Matrices
• 2.3 Matrix Addition and Scalar Multiplication
• 2.4 Summation Symbol
• 2.5 Matrix Multiplication
• 2.6 Transpose of a Matrix
• 2.7 Square Matrices
• 2.8 Powers of Matrices, Polynomials in Matrices
• 2.9 Invertible (Nonsingular) Matrices
• 2.10 Special Types of Square Matrices
• 2.11 Complex Matrices
• 2.12 Block Matrices
• Chapter 3 Systems of Linear Equations
• 3.1 Introduction
• 3.2 Basic Definitions, Solutions
• 3.3 Equivalent Systems, Elementary Operations
• 3.4 Small Square Systems of Linear Equations
• 3.5 Systems in Triangular and Echelon Forms
• 3.6 Gaussian Elimination
• 3.7 Echelon Matrices, Row Canonical Form, Row Equivalence
• 3.8 Gaussian Elimination, Matrix Formulation
• 3.9 Matrix Equation of a System of Linear Equations
• 3.10 Systems of Linear Equations and Linear Combinations of Vectors
• 3.11 Homogeneous Systems of Linear Equations
• 3.12 Elementary Matrices
• 3.13 𝐿𝑈 Decomposition
• Chapter 4 Vector Spaces
• 4.1 Introduction
• 4.2 Vector Spaces
• 4.3 Examples of Vector Spaces
• 4.4 Linear Combinations, Spanning Sets
• 4.5 Subspaces
• 4.6 Linear Spans, Row Space of a Matrix
• 4.7 Linear Dependence and Independence
• 4.8 Basis and Dimension
• 4.9 Application to Matrices, Rank of a Matrix
• 4.10 Sums and Direct Sums
• 4.11 Coordinates
• 4.12 Isomorphism of 𝑉 and 𝐾^𝑛
• 4.13 Full Rank Factorization
• 4.14 Generalized (Moore–Penrose) Inverse
• 4.15 Least-Square Solution
• Chapter 5 Linear Mappings
• 5.1 Introduction
• 5.2 Mappings, Functions
• 5.3 Linear Mappings (Linear Transformations)
• 5.4 Kernel and Image of a Linear Mapping
• 5.5 Singular and Nonsingular Linear Mappings, Isomorphisms
• 5.6 Operations with Linear Mappings
• 5.7 Algebra 𝐴(𝑉) of Linear Operators
• Chapter 6 Linear Mappings and Matrices
• 6.1 Introduction
• 6.2 Matrix Representation of a Linear Operator
• 6.3 Change of Basis
• 6.4 Similarity
• 6.5 Matrices and General Linear Mappings
• Chapter 7 Inner Product Spaces, Orthogonality
• 7.1 Introduction
• 7.2 Inner Product Spaces
• 7.3 Examples of Inner Product Spaces
• 7.4 Cauchy–Schwarz Inequality, Applications
• 7.5 Orthogonality
• 7.6 Orthogonal Sets and Bases
• 7.7 Gram–Schmidt Orthogonalization Process
• 7.8 Orthogonal and Positive Definite Matrices
• 7.9 Complex Inner Product Spaces
• 7.10 Normed Vector Spaces (Optional)
• Chapter 8 Determinants
• 8.1 Introduction
• 8.2 Determinants of Orders 1 and 2
• 8.3 Determinants of Order 3
• 8.4 Permutations
• 8.5 Determinants of Arbitrary Order
• 8.6 Properties of Determinants
• 8.7 Minors and Cofactors
• 8.8 Evaluation of Determinants
• 8.10 Applications to Linear Equations, Cramer’s Rule
• 8.11 Submatrices, Minors, Principal Minors
• 8.12 Block Matrices and Determinants
• 8.13 Determinants and Volume
• 8.14 Determinant of a Linear Operator
• 8.15 Multilinearity and Determinants
• Chapter 9 Diagonalization: Eigenvalues and Eigenvectors
• 9.1 Introduction
• 9.2 Polynomials of Matrices
• 9.3 Characteristic Polynomial, Cayley–Hamilton Theorem
• 9.4 Diagonalization, Eigenvalues and Eigenvectors
• 9.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices
• 9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms
• 9.7 Minimal Polynomial
• 9.8 Characteristic and Minimal Polynomials of Block Matrices
• Chapter 10 Canonical Forms
• 10.1 Introduction
• 10.2 Triangular Form
• 10.3 Invariance
• 10.4 Invariant Direct-Sum Decompositions
• 10.5 Primary Decomposition
• 10.6 Nilpotent Operators
• 10.7 Jordan Canonical Form
• 10.8 Cyclic Subspaces
• 10.9 Rational Canonical Form
• 10.10 Quotient Spaces
• Chapter 11 Linear Functionals and the Dual Space
• 11.1 Introduction
• 11.2 Linear Functionals and the Dual Space
• 11.3 Dual Basis
• 11.4 Second Dual Space
• 11.5 Annihilators
• 11.6 Transpose of a Linear Mapping
• Chapter 12 Bilinear, Quadratic, and Hermitian Forms
• 12.1 Introduction
• 12.2 Bilinear Forms
• 12.3 Bilinear Forms and Matrices
• 12.4 Alternating Bilinear Forms
• 12.5 Symmetric Bilinear Forms, Quadratic Forms
• 12.6 Real Symmetric Bilinear Forms, Law of Inertia
• 12.7 Hermitian Forms
• Chapter 13 Linear Operators on Inner Product Spaces
• 13.1 Introduction