 41.1: Explain why {(x, y) : x 5 y2} is not a function but {(x, y) : } is ...
 41.2: Can y 5 define a function from the set of positive integers to the ...
 41.3: In 35: a. Explain why each set of ordered pairs is or is not a func...
 41.4: In 35: a. Explain why each set of ordered pairs is or is not a func...
 41.5: In 35: a. Explain why each set of ordered pairs is or is not a func...
 41.6: In 611: a. Determine whether or not each graph represents a functio...
 41.7: In 611: a. Determine whether or not each graph represents a functio...
 41.8: In 611: a. Determine whether or not each graph represents a functio...
 41.9: In 611: a. Determine whether or not each graph represents a functio...
 41.10: In 611: a. Determine whether or not each graph represents a functio...
 41.11: In 611: a. Determine whether or not each graph represents a functio...
 41.12: In 1223, each set is a function from set A to set B. a. What is the...
 41.13: In 1223, each set is a function from set A to set B. a. What is the...
 41.14: In 1223, each set is a function from set A to set B. a. What is the...
 41.15: In 1223, each set is a function from set A to set B. a. What is the...
 41.16: In 1223, each set is a function from set A to set B. a. What is the...
 41.17: In 1223, each set is a function from set A to set B. a. What is the...
 41.18: In 1223, each set is a function from set A to set B. a. What is the...
 41.19: In 1223, each set is a function from set A to set B. a. What is the...
 41.20: In 1223, each set is a function from set A to set B. a. What is the...
 41.21: In 1223, each set is a function from set A to set B. a. What is the...
 41.22: In 1223, each set is a function from set A to set B. a. What is the...
 41.23: In 1223, each set is a function from set A to set B. a. What is the...
 41.24: The perimeter of a rectangle is 12 meters. a. If x is the length of...
 41.25: A candy store sells candy by the piece for 10 cents each. The amoun...
Solutions for Chapter 41: Relations And Functions
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 41: Relations And Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. Since 25 problems in chapter 41: Relations And Functions have been answered, more than 30706 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 41: Relations And Functions includes 25 full stepbystep solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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 nl  An).

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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.

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.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).