×
×

# Solutions for Chapter 67: Using Prime Factorization to Reduce Fractions

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

ISBN: 9781591417835

Solutions for Chapter 67: Using Prime Factorization to Reduce Fractions

Solutions for Chapter 67
4 5 0 316 Reviews
15
1
##### ISBN: 9781591417835

Since 30 problems in chapter 67: Using Prime Factorization to Reduce Fractions have been answered, more than 38882 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 67: Using Prime Factorization to Reduce Fractions includes 30 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Column space C (A) =

space of all combinations of the columns of A.

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• 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].

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Indefinite matrix.

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

• 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.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = 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.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Particular solution x p.

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

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Rank r (A)

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

• 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.

• 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.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

×