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Solutions for Chapter 4: Fitting a Line to Data

Discovering Algebra: An Investigative Approach | 2nd Edition | ISBN: 9781559537636 | Authors: Jerald Murdock, Ellen Kamischke, Eric Kamischke

Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition

ISBN: 9781559537636

Discovering Algebra: An Investigative Approach | 2nd Edition | ISBN: 9781559537636 | Authors: Jerald Murdock, Ellen Kamischke, Eric Kamischke

Solutions for Chapter 4: Fitting a Line to Data

Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. Since 11 problems in chapter 4: Fitting a Line to Data have been answered, more than 8469 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Fitting a Line to Data includes 11 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.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hankel matrix H.

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

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Indefinite matrix.

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

  • Length II x II.

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

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • 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).

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • Nullspace matrix N.

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

  • Pivot.

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

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

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • 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!

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • 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.

  • Symmetric matrix A.

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

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