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# Solutions for Chapter 7.1: Functions Defined on General Sets

## Full solutions for Discrete Mathematics with Applications | 4th Edition

ISBN: 9780495391326

Solutions for Chapter 7.1: Functions Defined on General Sets

Solutions for Chapter 7.1
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##### ISBN: 9780495391326

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 53 problems in chapter 7.1: Functions Defined on General Sets have been answered, more than 49058 students have viewed full step-by-step solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 7.1: Functions Defined on General Sets includes 53 full step-by-step solutions.

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

If diagonalizable, they share n eigenvectors.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• 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.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Graph G.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• 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.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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(DÂ» O.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• 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).

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).
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