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- Chapter 1:
- Chapter 10:
- Chapter 11:
- Chapter 12:
- Chapter 13:
- Chapter 14:
- Chapter 15:
- Chapter 16:
- Chapter 17:
- Chapter 18:
- Chapter 19:
- Chapter 2:
- Chapter 20:
- Chapter 21:
- Chapter 22:
- Chapter 23:
- Chapter 24:
- Chapter 25:
- Chapter 26:
- Chapter 27:
- Chapter 28:
- Chapter 29:
- Chapter 3:
- Chapter 30:
- Chapter 31:
- Chapter 32:
- Chapter 33:
- Chapter 4:
- Chapter 5:
- Chapter 6:
- Chapter 7:
- Chapter 8:
- Chapter 9:
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 22678 students have viewed full step-by-step answer.
Key Math Terms and definitions covered in this textbook
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.
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Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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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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Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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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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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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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.
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Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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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 •
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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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Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.