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Solutions for Chapter 6.3: Exponential Functions

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 6.3: Exponential Functions

Solutions for Chapter 6.3
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 6.3: Exponential Functions includes 126 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 126 problems in chapter 6.3: Exponential Functions have been answered, more than 34319 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Cholesky factorization

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

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • 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.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • Left nullspace N (AT).

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

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Semidefinite matrix A.

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

  • 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.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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