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# Solutions for Chapter 14: ELEMENTARY PROPERTIES

## Full solutions for Modern Algebra: An Introduction | 6th Edition

ISBN: 9780470384435

Solutions for Chapter 14: ELEMENTARY PROPERTIES

Solutions for Chapter 14
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##### ISBN: 9780470384435

This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 38 problems in chapter 14: ELEMENTARY PROPERTIES have been answered, more than 8967 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. Chapter 14: ELEMENTARY PROPERTIES includes 38 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• 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.

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.

• 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

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• 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.

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• 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.

• 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.

• 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.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• 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.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• 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.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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