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Solutions for Chapter 1.3: Models and Applications

College Algebra | 6th Edition | ISBN: 9780321782281 | Authors: Robert F. Blitzer

Full solutions for College Algebra | 6th Edition

ISBN: 9780321782281

College Algebra | 6th Edition | ISBN: 9780321782281 | Authors: Robert F. Blitzer

Solutions for Chapter 1.3: Models and Applications

Solutions for Chapter 1.3
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Textbook: College Algebra
Edition: 6
Author: Robert F. Blitzer
ISBN: 9780321782281

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.3: Models and Applications includes 86 full step-by-step solutions. College Algebra was written by and is associated to the ISBN: 9780321782281. This textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 86 problems in chapter 1.3: Models and Applications have been answered, more than 38869 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Cholesky factorization

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

  • 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.

  • 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!).

  • Gauss-Jordan method.

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

  • Left nullspace N (AT).

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

  • 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).

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Rank r (A)

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

  • Schwarz inequality

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

  • Semidefinite matrix A.

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

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Spanning set.

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

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Trace of A

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

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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