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# Solutions for Chapter 11.6: Events Involving Not and Or; Odds

## Full solutions for Thinking Mathematically | 6th Edition

ISBN: 9780321867322

Solutions for Chapter 11.6: Events Involving Not and Or; Odds

Solutions for Chapter 11.6
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##### ISBN: 9780321867322

Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Chapter 11.6: Events Involving Not and Or; Odds includes 106 full step-by-step solutions. Since 106 problems in chapter 11.6: Events Involving Not and Or; Odds have been answered, more than 70606 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

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

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

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

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

• Dimension of vector space

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

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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!).

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• 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.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Particular solution x p.

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

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

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