- 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: Age-Specific 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 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 4.11: 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; QR-Decomposition
- 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: LU-Decompositions
- 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
Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th Edition - Solutions by ChapterGet Full Solutions
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Invert A by row operations on [A I] to reach [I A-I].
Gram-Schmidt 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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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 •
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.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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.