- 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.)
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 edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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).
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(D» 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!
Constant down each diagonal = time-invariant (shift-invariant) filter.
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.