- 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
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Every v in V is orthogonal to every w in W.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).