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- Chapter 18:
- Chapter 19:
- Chapter 2:
- Chapter 20:
- Chapter 21:
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Contemporary Abstract Algebra 8th Edition - Solutions by Chapter
Full solutions for Contemporary Abstract Algebra | 8th Edition
ISBN: 9781133599708
Solutions by Chapter
This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. This expansive textbook survival guide covers the following chapters: 34. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. The full step-by-step solution to problem in Contemporary Abstract Algebra were answered by , our top Math solution expert on 07/25/17, 05:55AM. Since problems from 34 chapters in Contemporary Abstract Algebra have been answered, more than 85592 students have viewed full step-by-step answer.
Key Math Terms and definitions covered in this textbook
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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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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Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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Orthogonal subspaces.
Every v in V is orthogonal to every w in W.
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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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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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Rotation matrix
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.