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Solutions for Chapter 9.5: Parametric Equations

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 9.5: Parametric Equations

Solutions for Chapter 9.5
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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 expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 9.5: Parametric Equations includes 68 full step-by-step solutions. Since 68 problems in chapter 9.5: Parametric Equations have been answered, more than 33339 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Cofactor Cij.

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

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • 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.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • 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).

  • 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).

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(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.

  • 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 ().

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

    Column vectors by convention.

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Standard basis for Rn.

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

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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