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Solutions for Chapter 4.1: Exponential Functions

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Full solutions for College Algebra | 7th Edition

ISBN: 9780134469164

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Solutions for Chapter 4.1: Exponential Functions

Solutions for Chapter 4.1
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Textbook: College Algebra
Edition: 7
Author: Robert F. Blitzer
ISBN: 9780134469164

This textbook survival guide was created for the textbook: College Algebra , edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780134469164. Chapter 4.1: Exponential Functions includes 99 full step-by-step solutions. Since 99 problems in chapter 4.1: Exponential Functions have been answered, more than 32729 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.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • 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.

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • 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.

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Kronecker product (tensor product) A ® B.

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

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Pivot.

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

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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