- 19-1.1: Find the annual premium for a 10-year level term insurance policy w...
- 19-1.2: Find the annual premium for a whole-life insurance policy with a fa...
- 19-1.3: What are the quarterly payments on a $100,000 whole-life insurance ...
- 19-1.4: What are the monthly payments on a $200,000 universal-life insuranc...
- 19-1.5: Compare the premiums for a 10-year level term policy for $75,000 fo...
- 19-1.6: Compare the premiums for a 20-year level term policy for $500,000 f...
- 19-1.7: Compare the annual life insurance premium of Jenny Davis who is 35 ...
- 19-1.8: Compare the annual life insurance premium of Garrett Townse who is ...
- 19-1.9: Cindy Franklin started a whole-life insurance policy for $250,000 w...
- 19-1.10: Parker Waters $200,000 whole-life insurance policy has a cash value...
Solutions for Chapter 19-1: LIFE INSURANCE
Full solutions for Business Math, | 9th Edition
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).
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
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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!).
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.