- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Remove row i and column j; multiply the determinant by (-I)i + j •
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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