 44.1: If the domain of the function f(x) 5 3 2 x is the set of real numbe...
 44.2: Eric said that if f(x) 5 2 2 x, then y 5 2 2 x when x # 2 and y 5 x...
 44.3: In 36, find the coordinates of the ordered pair with the smallest v...
 44.4: In 36, find the coordinates of the ordered pair with the smallest v...
 44.5: In 36, find the coordinates of the ordered pair with the smallest v...
 44.6: In 36, find the coordinates of the ordered pair with the smallest v...
 44.7: In 710, the domain of each function is the set of real numbers. a. ...
 44.8: In 710, the domain of each function is the set of real numbers. a. ...
 44.9: In 710, the domain of each function is the set of real numbers. a. ...
 44.10: In 710, the domain of each function is the set of real numbers. a. ...
 44.11: A(2, 7) is a fixed point in the coordinate plane. Let B(x, 7) be an...
 44.12: Along the New York State Thruway there are distance markers that gi...
 44.13: a. Sketch the graph of y 5 x. b. Sketch the graph of y 5 x 1 2 c. S...
 44.14: a. Sketch the graph of y 5 x. b. Sketch the graph of y 5 x 1 2. c. ...
 44.15: a. Sketch the graph of y 5 x. b. Sketch the graph of y 5 2x. c. Des...
 44.16: a. Sketch the graph of y 5 x. b. Sketch the graph of y 5 2x. c. Ske...
 44.17: a. Draw the graphs of y 5 x 1 3 and y 5 5. b. From the graph drawn ...
 44.18: a. Draw the graphs of y 5 2x 2 4 and y 5 22. b. From the graph draw...
Solutions for Chapter 44: Absolute Value Functions
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 44: Absolute Value Functions
Get Full SolutionsSince 18 problems in chapter 44: Absolute Value Functions have been answered, more than 29458 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This 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. Chapter 44: Absolute Value Functions includes 18 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.