- Chapter 1: Linear Equations and Matrices
- Chapter 1.1: Systems of Linear Equations
- Chapter 1.2: Matrices
- Chapter 1.3: Matrix Multiplication
- Chapter 1.4: Algebraic Properties of Matrix Operations
- Chapter 1.5: Special Types of Matrices and Partitioned Matrices
- Chapter 1.6: Matrix Transformations
- Chapter 1.7: Computer Graphics (Optional)
- Chapter 1.8: Correlation Coefficient (Optional)
- Chapter 2: Solving linear Systems
- Chapter 2.1: Echelon Form of a Matrix
- Chapter 2.2: Solving Lint!ar Systt!ms
- Chapter 2.3: Elementary Matrices; Finding A -
- Chapter 2.4: Equivalent Matrices
- Chapter 2.5: LV-Factorization (Optional)
- Chapter 3: Determinants
- Chapter 3.1: Definition
- Chapter 3.2: Properties of Determinants
- Chapter 3.3: Cofactor Expansion
- Chapter 3.4: Inverse of a Matrix
- Chapter 3.5: Other Applications of Determinants
- Chapter 3.6: Determinants from a Computational Point of View
- Chapter 4: Real Vector Spaces
- Chapter 4.1: Vectors in the Plane and in 3-Space
- Chapter 4.2: Vector Spaces
- Chapter 4.3: Subspaces
- Chapter 4.4: Span
- Chapter 4.5: Linear Independence
- Chapter 4.6: Basis and Dimension
- Chapter 4.7: Homogeneous Systems
- Chapter 4.8: Coordinates and Isomorphisms
- Chapter 4.9: Rank of a Matrix
- Chapter 5: Inner Product Spaces
- Chapter 5.1: Length and Direction in R2 and R3
- Chapter 5.2: Cross Product in RJ (Optional)
- Chapter 5.3: Inner Product Spaces
- Chapter 5.4: Gram*-Schmidtt Process
- Chapter 5.5: Orthogonal Complements
- Chapter 5.6: Least Squares (Optional)
- Chapter 6: Li near Transformations and Matrices
- Chapter 6.1: Definition and Examples
- Chapter 6.2: Kernel and Range of a Linear Transformation
- Chapter 6.3: Matrix of a Linear Transformation
- Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)
- Chapter 6.5: Similarity
- Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)
- Chapter 7: Eigenvalues and Eigenvectors
- Chapter 7.1: Eigenvalues and Eigenvectors
- Chapter 7.2: Diagonalization and Similar Matrices
- Chapter 7.3: Diagonalization of Symmetric Matrices
- Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
- Chapter 8.2: Spectral Decomposition and Singular Value Decomposition
- Chapter 8.3: Dominant Eigenvalue and Principal Component Analysis
- Chapter 8.4: Differential Equations
- Chapter 8.6: Real Quadratic Forms
- Chapter 8.7: Conic Sections
- Chapter 8.8: Quadric Surfaces
Elementary Linear Algebra with Applications 9th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
ISBN: 9780132296540
Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This expansive textbook survival guide covers the following chapters: 57. Since problems from 57 chapters in Elementary Linear Algebra with Applications have been answered, more than 58343 students have viewed full step-by-step answer. The full step-by-step solution to problem in Elementary Linear Algebra with Applications were answered by , our top Math solution expert on 01/30/18, 04:18PM. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9.
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
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Block matrix.
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.
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Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
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Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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Condition number
cond(A) = c(A) = IIAIlIIA-III = 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.
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Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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Diagonalization
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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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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Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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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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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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Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
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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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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.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
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Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).