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Solutions for Chapter 7.3: The Hyperbola

College Algebra: Graphs and Models | 5th Edition | ISBN: 9780321783950 | Authors: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Full solutions for College Algebra: Graphs and Models | 5th Edition

ISBN: 9780321783950

College Algebra: Graphs and Models | 5th Edition | ISBN: 9780321783950 | Authors: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Solutions for Chapter 7.3: The Hyperbola

Solutions for Chapter 7.3
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Textbook: College Algebra: Graphs and Models
Edition: 5
Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna
ISBN: 9780321783950

This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 53 problems in chapter 7.3: The Hyperbola have been answered, more than 29552 students have viewed full step-by-step solutions from this chapter. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. Chapter 7.3: The Hyperbola includes 53 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • 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!

  • 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.)

  • Cholesky factorization

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

  • Circulant matrix C.

    Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

  • Cofactor Cij.

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

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • 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.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Rank r (A)

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

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric matrix A.

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

  • Vector space V.

    Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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