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Solutions for Chapter 2.3: More on Solving Linear Equations

Beginning Algebra | 11th Edition | ISBN: 9780321673480 | Authors: Margaret L. Lial John Hornsby, Terry McGinnis

Full solutions for Beginning Algebra | 11th Edition

ISBN: 9780321673480

Beginning Algebra | 11th Edition | ISBN: 9780321673480 | Authors: Margaret L. Lial John Hornsby, Terry McGinnis

Solutions for Chapter 2.3: More on Solving Linear Equations

Solutions for Chapter 2.3
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Textbook: Beginning Algebra
Edition: 11
Author: Margaret L. Lial John Hornsby, Terry McGinnis
ISBN: 9780321673480

Since 86 problems in chapter 2.3: More on Solving Linear Equations have been answered, more than 40137 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.3: More on Solving Linear Equations includes 86 full step-by-step solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • 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.)

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • 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.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • 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.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • 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).

  • Partial pivoting.

    In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Similar matrices A and B.

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

  • Skew-symmetric matrix K.

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

  • Toeplitz matrix.

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

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • 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·

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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