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Solutions for Chapter 9-1: Exponential Functions

Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition | ISBN: 9780078738302 | Authors: McGraw-Hill Education

Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition

ISBN: 9780078738302

Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition | ISBN: 9780078738302 | Authors: McGraw-Hill Education

Solutions for Chapter 9-1: Exponential Functions

Solutions for Chapter 9-1
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Textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2)
Edition: 1
Author: McGraw-Hill Education
ISBN: 9780078738302

Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 9-1: Exponential Functions includes 79 full step-by-step solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 79 problems in chapter 9-1: Exponential Functions have been answered, more than 53704 students have viewed full step-by-step solutions from this chapter.

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

    Tv = Av + Vo = linear transformation plus shift.

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • 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.

  • 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.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Left inverse A+.

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

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

    Vector addition and scalar multiplication.

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • 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.

  • 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.

  • 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 •

  • 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(D» O.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row 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.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

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