- 13-2.1: Find the amount that should be set aside today to yield the desired...
- 13-2.2: Find the amount that should be set aside today to yield the desired...
- 13-2.3: Find the amount that should be set aside today to yield the desired...
- 13-2.4: Find the amount that should be set aside today to yield the desired...
- 13-2.5: Compute the amount of money to be set aside today to ensure a futur...
- 13-2.6: How much should Linda Bryan set aside now to buy equipment that cos...
- 13-2.7: Ronnie Cox has just inherited $27,000. How much of this money shoul...
- 13-2.8: Shirley Riddle received a $10,000 gift from her mother and plans a ...
- 13-2.9: Rosa Burnett needs $2,000 in three years to make the down payment o...
- 13-2.10: Use Table 13-3 to calculate the amount of money that must be invest...
- 13-2.11: Dewey Sykes plans to open a business in four years when he retires....
- 13-2.12: Charlie Bryant has a child who will be college age in five years. H...
Solutions for Chapter 13-2: PRESENT VALUE
Full solutions for Business Math, | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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 n-l - An).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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.
Gram-Schmidt 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.
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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Every v in V is orthogonal to every w in W.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.