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Solutions for Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2

Saxon Math, Course 1 | 1st Edition | ISBN: 9781591417835 | Authors: Stephan Hake

Full solutions for Saxon Math, Course 1 | 1st Edition

ISBN: 9781591417835

Saxon Math, Course 1 | 1st Edition | ISBN: 9781591417835 | Authors: Stephan Hake

Solutions for Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2

Since 30 problems in chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2 have been answered, more than 36085 students have viewed full step-by-step solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2 includes 30 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

Key Math Terms and definitions covered in this textbook
  • Cholesky factorization

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

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Length II x II.

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

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Multiplication Ax

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

  • 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.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Nullspace matrix N.

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

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • 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.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • 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.

  • 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.

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