 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 Matrices
 Chapter 1.7: Diagonal,Triangular, and Symmetric Matrices
 Chapter 1.8: MatrixTransformations
 Chapter 1.9: Applications of Linear Systems
 Chapter 10.1: Constructing Curves and SurfacesThrough Specified Points
 Chapter 10.11: ComputedTomography
 Chapter 10.12: Fractals
 Chapter 10.13: Chaos
 Chapter 10.14: Cryptography
 Chapter 10.15: Genetics
 Chapter 10.16: AgeSpecific Population Growth
 Chapter 10.17: Harvesting of Animal Populations
 Chapter 10.18: A Least Squares Model for Human Hearing
 Chapter 10.19: Warps and Morphs
 Chapter 10.2: The Earliest Applications of Linear Algebra
 Chapter 10.3: ubic Spline Interpolation
 Chapter 10.4: Markov Chains
 Chapter 10.5: GraphTheory
 Chapter 10.6: Games of Strategy
 Chapter 10.7: Leontief Economic Models
 Chapter 10.8: Forest Management
 Chapter 10.9: Computer Graphics
 Chapter 2: Determinants
 Chapter 2.1: Determinants by Cofactor Expansion
 Chapter 2.2: Evaluating Determinants by Row Reduction
 Chapter 2.3: Properties of Determinants; Cramers 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 4.11: Geometry of Matrix Operators on R2
 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: Basic Matrix Transformations in R2 and R3
 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 5.5: Dynamical Systems and Markov Chains
 Chapter 6: Inner Product Spaces
 Chapter 6.1: Inner Products
 Chapter 6.2: Angle and Orthogonality in Inner Product Spaces
 Chapter 6.3: GramSchmidt Process; QRDecomposition
 Chapter 6.4: Best Approximation; Least Squares
 Chapter 6.5: Mathematical Modeling Using Least Squares
 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: General Linear Transformations
 Chapter 8.1: General Linear Transformations
 Chapter 8.2: Compositions and InverseTransformations
 Chapter 8.3: Isomorphism
 Chapter 8.4: Matrices for General LinearTransformations
 Chapter 8.5: Similarity
 Chapter 9: Numerical Methods
 Chapter 9.1: LUDecompositions
 Chapter 9.2: The Power Method
 Chapter 9.3: Comparison of Procedures for Solving Linear Systems
 Chapter 9.4: Singular Value Decomposition
 Chapter 9.5: Data Compression Using Singular Value Decomposition
Elementary Linear Algebra, Binder Ready Version: Applications Version 11th Edition  Solutions by Chapter
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition  Solutions by Chapter
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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