- 73.1: Tomass temperature was 102F. Normal body temperatureis 98.6F. How m...
- 73.2: Jill has read 42 pages of a 180-page book. How manypages are left f...
- 73.3: If Jill wants to finish the book in the next three days, thenshe sh...
- 73.4: Write 2.5 as a reduced mixed number.
- 73.5: Write 0.35 as a reduced fraction.
- 73.6: What is the total cost of a $12.60 item when 712% (0.075) sales tax...
- 73.7: 34 2 113
- 73.8: (100 102 ) 52
- 73.9: 3 213 134
- 73.10: 516 312
- 73.11: 34 112
- 73.12: 7 0.4
- 73.13: Compare: a. 52 25 b. 0.3 0.125
- 73.14: The diameter of a quarter is about 2.4 cm. a. What is the circumfer...
- 73.15: Find each unknown number:25m = 0.175
- 73.16: Find each unknown number:1.2 + y + 4.25 = 7
- 73.17: Which digit is in the ten-thousands place in 123,456.78?
- 73.18: Arrange these numbers in order from least to greatest:1, 12, 110,14, 0
- 73.19: Write the prime factorization of 200 using exponents.
- 73.20: The store offered a 20% discount on all tools. The regular price of...
- 73.21: The length of AB is 16 mm. The length of AC is 50 mm. Whatis the le...
- 73.22: One half of the area of this square is shaded.What is the area of t...
- 73.23: Is every square a rectangle?
- 73.24: 22 232
- 73.25: The fractions chart from Lesson 72 says that the propershape for mu...
- 73.26: Refer to this coordinate plane to answer problems 26 and 27.Identif...
- 73.27: Refer to this coordinate plane to answer problems 26 and 27.Name th...
- 73.28: If s equals 9, what does s2 equal?
- 73.29: Name an every day object that has the same shape as each of thesege...
- 73.30: The measure of W inparallelogram WXYZ is 75. a. What is the measure...
Solutions for Chapter 73: Exponents Writing Decimal Numbers as Fractions, Part 2
Full solutions for Saxon Math, Course 1 | 1st Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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).
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Solvable system Ax = b.
The right side b is in the column space of A.