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# Solutions for Chapter 8.6: Parametric Equations, Graphs, and Applications

## Full solutions for Trigonometry | 10th Edition

ISBN: 9780321671776

Solutions for Chapter 8.6: Parametric Equations, Graphs, and Applications

Solutions for Chapter 8.6
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##### ISBN: 9780321671776

Chapter 8.6: Parametric Equations, Graphs, and Applications includes 56 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 8.6: Parametric Equations, Graphs, and Applications have been answered, more than 32728 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Trigonometry was written by and is associated to the ISBN: 9780321671776.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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).

• 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.

• 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.

• 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.

• Hankel matrix H.

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

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• 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.

• 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 •

• Outer product uv T

= column times row = rank one matrix.

• Particular solution x p.

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

• 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.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

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

P = aaT laTa has rank l.

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• 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.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

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