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Solutions for Chapter 11.1: The Fundamental Counting Principle

Thinking Mathematically | 6th Edition | ISBN: 9780321867322 | Authors: Robert F. Blitzer

Full solutions for Thinking Mathematically | 6th Edition

ISBN: 9780321867322

Thinking Mathematically | 6th Edition | ISBN: 9780321867322 | Authors: Robert F. Blitzer

Solutions for Chapter 11.1: The Fundamental Counting Principle

Solutions for Chapter 11.1
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Textbook: Thinking Mathematically
Edition: 6
Author: Robert F. Blitzer
ISBN: 9780321867322

Chapter 11.1: The Fundamental Counting Principle includes 28 full step-by-step solutions. Since 28 problems in chapter 11.1: The Fundamental Counting Principle have been answered, more than 64711 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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

  • 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.

  • Eigenvalue A and eigenvector x.

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

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • Left nullspace N (AT).

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

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • 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.

  • 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.

  • Schwarz inequality

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

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

  • Toeplitz matrix.

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

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

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