- 9.1: Mr. Washington shows his class a pattern of equilateral triangles. ...
- 9.2: Tanya made the data table below. She plans to graph the line on a g...
- 9.3: A bicycle courier in New York City records the distance she travels...
- 9.4: What is the slope of the line? O -2 -3 -4 -4-3-2 1 4 2 3 1 2 3 4 y ...
- 9.5: The scatter plot below shows the relationship between the number of...
- 9.6: Shannon and Chris sell magazines to raise money for their school. F...
- 9.7: What is the value of the function f(x) = 3x - 1 when x = 2? A 3 B 5...
Solutions for Chapter 9: Math Connects: Concepts, Skills, and Problem Solving Course 3 0th Edition
Full solutions for Math Connects: Concepts, Skills, and Problem Solving Course 3 | 0th Edition
Tv = Av + Vo = linear transformation plus shift.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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).
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).