- Chapter 1: Linear Equations
- Chapter 1.1: Introduction to Linear Systems
- Chapter 1.2: Matrices, Vectors, and Gauss-Jordan Elimination
- Chapter 1.3: On the Solutions of Linear Systems; Matrix Algebra
- Chapter 2: Linear Transformations
- Chapter 2.1: Introduction to Linear Transformations and Their Inverses
- Chapter 2.2: Linear Transformations in Geometry
- Chapter 2.3: Matrix Products
- Chapter 2.4: The Inverse of a Linear Transformation
- Chapter 3.1: Image and Kernel of a Linear Transformation
- Chapter 3.2: Subspaces of R"; Bases and Linear Independence
- Chapter 3.3: The Dimension of a Subspace of R"
- Chapter 3.4: Coordinates
- Chapter 4: Linear Spaces
- Chapter 4.1: Introduction to Linear Spaces
- Chapter 4.2: Linear Transformations and Isomorphisms
- Chapter 4.3: Th e Matrix of a Linear Transformation
- Chapter 5: Orthogonality and Least Squares
- Chapter 5.1: Orthogonal Projections and Orthonormal Bases
- Chapter 5.2: Gram-Schmidt Process and QR Factorization
- Chapter 5.3: Orthogonal Transformations and Orthogonal Matrices
- Chapter 5.4: Least Squares and Data Fitting
- Chapter 5.5: Inner Product Spaces
- Chapter 6: Determinants
- Chapter 6.1: Introduction to Determinants
- Chapter 6.2: Properties of the Determinant
- Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule
- Chapter 7: Eigenvalues and Eigenvectors
- Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example
- Chapter 7.2: Finding the Eigenvalues of a Matrix
- Chapter 7.3: Finding the Eigenvectors of a Matrix
- Chapter 7.4: Diagonalization
- Chapter 7.5: Complex Eigenvalues
- Chapter 7.6: Stability
- Chapter 8: Symmetric Matrices and Quadratic Forms
- Chapter 8.1: Symmetric Matrices
- Chapter 8.2: Quadratic Forms
- Chapter 8.3: Singular Values
- Chapter 9.1: An Introduction to Continuous Dynamical Systems
- Chapter 9.2: The Complex Case: Eulers Formula
- Chapter 9.3: Linear Differential Operators and Linear Differential Equations
Linear Algebra with Applications 4th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 4th Edition
ISBN: 9780136009269
Linear Algebra with Applications | 4th Edition - Solutions by Chapter
Get Full SolutionsThe full step-by-step solution to problem in Linear Algebra with Applications were answered by , our top Math solution expert on 03/15/18, 05:20PM. Since problems from 41 chapters in Linear Algebra with Applications have been answered, more than 7068 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters: 41. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269.
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
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Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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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.
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Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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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.
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Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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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.
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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.
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Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.
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Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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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.