 161.1: Find the indicated amounts for the fixedrate mortgages. $100,000 $...
 161.2: Find the indicated amounts for the fixedrate mortgages. $183,000 $...
 161.3: Find the indicated amounts for the fixedrate mortgages. $95,000 $8...
 161.4: Find the indicated amounts for the fixedrate mortgages. $125,500 2...
 161.5: Find the indicated amounts for the fixedrate mortgages. $495,750 1...
 161.6: Find the indicated amounts for the fixedrate mortgages. $83,750 15...
 161.7: Stephen Black has just purchased a home for $155,000. Northridge Mo...
 161.8: Find the total interest Stephen will pay if he pays the loan on sch...
 161.9: If Stephen made the same loan for 20 years, how much interest would...
 161.10: How much would Stephens monthly payment increase for a 20year mort...
 161.11: The annual insurance premium on Maria Snyders home is $2,074 and th...
 161.12: Susan Blair has a 25year home mortgage of $208,917 at 4.75% intere...
 161.13: Use the formula or a calculator application to find the monthly pay...
 161.14: Use the formula or a calculator application to find the monthly pay...
Solutions for Chapter 161: MORTGAGE PAYMENTS
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 161: MORTGAGE PAYMENTS
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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 •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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