- Chapter 1: Systems of Linear Equations and Matrices
- Chapter 1.1: Introduction to Systems of Linear Equations
- Chapter 1.2: Gaussian Elimination
- Chapter 1.3: Matrices and Matrix Operations
- Chapter 1.4: Inverses; Algebraic Properties of Matrices
- Chapter 1.5: Elementary Matrices and a Method for Finding A1
- Chapter 1.6: More on Linear Systems and Invertible Matrics
- Chapter 1.7: Diagonal, Triangular, and Symmetric Matrices
- Chapter 1.8: Applications of Linear Systems
- Chapter 1.9: Leontief Input-Output Models
- Chapter 10.1: Constructing Curves and Surfaces Through Specified Points
- Chapter 10.10: Computer Graphics
- Chapter 10.11: Equilibrium Temperature Distributions
- Chapter 10.12: Computed Tomography
- Chapter 10.13: Fractals
- Chapter 10.14: Chaos
- Chapter 10.15: Cryptography
- Chapter 10.16: Genetics
- Chapter 10.17: Age-Specific Population Growth
- Chapter 10.18: Harvesting of Animal Populations
- Chapter 10.19: A Least Squares Model for Human Hearing
- Chapter 10.2: Geometric Linear Programming
- Chapter 10.20: Warps and Morphs
- Chapter 10.3: The Earliest Applications of Linear Algebra
- Chapter 10.4: Cubic Spline Interpolation
- Chapter 10.5: Markov Chains
- Chapter 10.6: Graph Theory
- Chapter 10.7: Games of Strategy
- Chapter 10.8: Leontief Economic Models
- Chapter 10.9: Forest Management
- Chapter 2: Determinants
- Chapter 2.1: Determinants by Cofactor Expansion
- Chapter 2.2: Evaluating Determinants by Row Reduction
- Chapter 2.3: Properties of Determinants; Cramer's Rule
- Chapter 3: Euclidean Vector Spaces
- Chapter 3.1: Vectors in 2-Space, 3-Space, and n-Space
- Chapter 3.2: Norm, Dot Product, and Distance in Rn
- Chapter 3.3: Orthogonality
- Chapter 3.4: The Geometry of Linear Systems
- Chapter 3.5: Cross Product
- Chapter 4: General Vector Spaces
- Chapter 4.1: Real Vector Spaces
- Chapter 4.10: Properties of Matrix Transformations
- Chapter 4.11: Geometry of Matrix Operators on
- Chapter 4.12: Dynamical Systems and Markov Chains
- Chapter 4.2: Subspaces
- Chapter 4.3: Linear Independence
- Chapter 4.4: Coordinates and Basis
- Chapter 4.5: Dimension
- Chapter 4.6: Change of Basis
- Chapter 4.7: Row Space, Column Space, and Null Space
- Chapter 4.8: Rank, Nullity, and the Fundamental Matrix Spaces
- Chapter 4.9: Matrix Transformations from Rn to Rm
- Chapter 5: Eigenvalues and Eigenvectors
- Chapter 5.1: Eigenvalues and Eigenvectors
- Chapter 5.2: Diagonalization
- Chapter 5.3: Complex Vector Spaces
- Chapter 5.4: Differential Equations
- Chapter 6: Inner Product Spaces
- Chapter 6.1: Inner Products
- Chapter 6.2: Inner Products
- Chapter 6.3: GramSchmidt Process; QR-Decomposition
- Chapter 6.4: Best Approximation; Least Squares
- Chapter 6.5: Least Squares Fitting to Data
- Chapter 6.6: Function Approximation; Fourier Series
- Chapter 7: Diagonalization and Quadratic Forms
- Chapter 7.1: Orthogonal Matrices
- Chapter 7.2: Orthogonal Diagonalization
- Chapter 7.3: Quadratic Forms
- Chapter 7.4: Optimization Using Quadratic Forms
- Chapter 7.5: Hermitian, Unitary, and Normal Matrices
- Chapter 8: Linear Transformation
- Chapter 8.1: General Linear Transformations
- Chapter 8.2: Isomorphism
- Chapter 8.3: Compositions and Inverse Transformations
- Chapter 8.4: Matrices for General Linear Transformations
- Chapter 8.5: Similarity
- Chapter 9: Numerical Methods
- Chapter 9.1: LU-Decompositions
- Chapter 9.2: The Power Method
- Chapter 9.3: Internet Search Engines
- Chapter 9.4: Comparison of Procedures for Solving Linear Systems
- Chapter 9.5: Singular Value Decomposition
Elementary Linear Algebra: Applications Version 10th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Elementary Linear Algebra: Applications Version | 10th Edition - Solutions by ChapterGet Full Solutions
Tv = Av + Vo = linear transformation plus shift.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Invert A by row operations on [A I] to reach [I A-I].
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
A directed graph that has constants Cl, ... , Cm associated with the edges.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.