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Solutions for Chapter 2.4: Complex Numbers

Precalculus With Limits A Graphing Approach | 5th Edition | ISBN: 9780618851522 | Authors: Ron Larson Robert Hostetler, Bruce H. Edwards, David C. Falvo (Contributor)

Full solutions for Precalculus With Limits A Graphing Approach | 5th Edition

ISBN: 9780618851522

Precalculus With Limits A Graphing Approach | 5th Edition | ISBN: 9780618851522 | Authors: Ron Larson Robert Hostetler, Bruce H. Edwards, David C. Falvo (Contributor)

Solutions for Chapter 2.4: Complex Numbers

Solutions for Chapter 2.4
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Textbook: Precalculus With Limits A Graphing Approach
Edition: 5
Author: Ron Larson Robert Hostetler, Bruce H. Edwards, David C. Falvo (Contributor)
ISBN: 9780618851522

This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 90 problems in chapter 2.4: Complex Numbers have been answered, more than 32478 students have viewed full step-by-step solutions from this chapter. Chapter 2.4: Complex Numbers includes 90 full step-by-step solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Key Math Terms and definitions covered in this textbook
  • 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].

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

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

    Use AT for complex A.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • 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.

  • 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).

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Nullspace matrix N.

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

  • 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.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • 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).

  • 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 picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • 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.

  • Similar matrices A and B.

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

  • 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·

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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