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# Solutions for Chapter 10.4: The Hyperbola

## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Solutions for Chapter 10.4: The Hyperbola

Solutions for Chapter 10.4
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##### ISBN: 9780132854351

Chapter 10.4: The Hyperbola includes 86 full step-by-step solutions. Since 86 problems in chapter 10.4: The Hyperbola have been answered, more than 56134 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Column space C (A) =

space of all combinations of the columns of A.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Length II x II.

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

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• 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.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.