- 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 Solutions
Solutions by Chapter
The 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 12203 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.
Key Math Terms and definitions covered in this textbook
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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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].
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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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Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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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).
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Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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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).
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.