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Solutions for Chapter 5: College Algebra and Trigonometry 7th Edition

College Algebra and Trigonometry | 7th Edition | ISBN: 9781439048603 | Authors: Richard N. Aufmann

Full solutions for College Algebra and Trigonometry | 7th Edition

ISBN: 9781439048603

College Algebra and Trigonometry | 7th Edition | ISBN: 9781439048603 | Authors: Richard N. Aufmann

Solutions for Chapter 5

Solutions for Chapter 5
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Textbook: College Algebra and Trigonometry
Edition: 7
Author: Richard N. Aufmann
ISBN: 9781439048603

Since 76 problems in chapter 5 have been answered, more than 5945 students have viewed full step-by-step solutions from this chapter. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Chapter 5 includes 76 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

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

    Tv = Av + Vo = linear transformation plus shift.

  • 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.

  • 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.

  • 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

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • 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.

  • Gauss-Jordan method.

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

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Hypercube matrix pl.

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

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • 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.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Outer product uv T

    = column times row = rank one matrix.

  • 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.

  • Spanning set.

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

  • Toeplitz matrix.

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

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