 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 InputOutput 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: AgeSpecific 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 2Space, 3Space, and nSpace
 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; QRDecomposition
 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: LUDecompositions
 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
ISBN: 9780470432051
Elementary Linear Algebra: Applications Version  10th Edition  Solutions by Chapter
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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.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.