 10.6.1E: Consider the tree shown below with root a.a. What is the level of n...
 10.6.2E: Consider the tree shown below with root v0.a. What is the level of ...
 10.6.3E: Draw binary trees to represent the following expressions:a. a • b –...
 10.6.4E: In each of either draw a graph with the given specifications or exp...
 10.6.5E: In each of either draw a graph with the given specifications or exp...
 10.6.6E: In each of either draw a graph with the given specifications or exp...
 10.6.7E: In each of either draw a graph with the given specifications or exp...
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 10.6.9E: In each of either draw a graph with the given specifications or exp...
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 10.6.11E: In each of either draw a graph with the given specifications or exp...
 10.6.12E: In each of either draw a graph with the given specifications or exp...
 10.6.13E: In each of either draw a graph with the given specifications or exp...
 10.6.14E: In each of either draw a graph with the given specifications or exp...
 10.6.15E: In each of either draw a graph with the given specifications or exp...
 10.6.16E: In each of either draw a graph with the given specifications or exp...
 10.6.17E: In each of either draw a graph with the given specifications or exp...
 10.6.18E: In each of either draw a graph with the given specifications or exp...
 10.6.19E: In each of either draw a graph with the given specifications or exp...
 10.6.20E: In each of either draw a graph with the given specifications or exp...
Solutions for Chapter 10.6: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.6
Get Full SolutionsDiscrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Since 20 problems in chapter 10.6 have been answered, more than 23942 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.6 includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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 nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

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.

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 •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.
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