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# Solutions for Chapter 10.7: Discrete Mathematics with Applications 4th Edition

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

ISBN: 9780495391326

Solutions for Chapter 10.7

Solutions for Chapter 10.7
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##### ISBN: 9780495391326

Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 27 problems in chapter 10.7 have been answered, more than 52796 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.7 includes 27 full step-by-step solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Key Math Terms and definitions covered in this textbook
• Column space C (A) =

space of all combinations of the columns of A.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• 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.

• 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].

• 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

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Linear combination cv + d w or L C jV j.

• 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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthogonal subspaces.

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

• 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.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• 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.

• 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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