- 94.1: Ten of the thirty students on the bus were boys. What percentof the...
- 94.2: On the Celsius scale water freezes at 0C and boils at100C. What tem...
- 94.3: If the length of segment AB is 13 the length of segment AC,and if s...
- 94.4: What percent of this group is shaded?
- 94.5: Change 1 23 to a percent by multiplying 1 23 by 100%.
- 94.6: Change 1.5 to a percent by multiplying 1.5 by 100%.
- 94.7: 6.4 614 (Begin by writing 614 as a decimal number.)
- 94.8: 104 103
- 94.9: How much is 34 of 360?
- 94.10: Tommy placed a cylindrical can of spaghetti sauceon the counter. He...
- 94.11: Tommy placed a cylindrical can of spaghetti sauceon the counter. He...
- 94.12: 312 134 458
- 94.13: 910 56 89
- 94.14: Write 250,000 in expanded notation using exponents.
- 94.15: $8.47 + 95 + $12
- 94.16: 37.5 100
- 94.17: 37 21x
- 94.18: 33 , 13 100
- 94.19: If ninety percent of the answers were correct, then what percent we...
- 94.20: Write the decimal number one hundred twenty and threehundredths.
- 94.21: Arrange these numbers in order from least to greatest:-2.5, 25,52 5.2
- 94.22: A pyramid with a square base has how many edges?
- 94.23: What is the area of this parallelogram?
- 94.24: The parallelogram in problem 23 is divided into twocongruent triang...
- 94.25: During the year, the temperature ranged from 37F in winter to 103Fi...
- 94.26: The coordinates of the three vertices of a triangle are (0, 0),(0, ...
- 94.27: Margies first nine test scores are shown below.21, 25, 22, 19, 22, ...
- 94.28: 23 225 3 42 24
- 94.29: Sandra filled the aquarium with 24 quarts of water. How many gallon...
- 94.30: A bag contains lettered tiles, two for each letter of the alphabet....
Solutions for Chapter 94: Writing Fractions and Decimals as Percents, Part 2
Full solutions for Saxon Math, Course 1 | 1st Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.)
A = CTC = (L.J]))(L.J]))T for positive definite A.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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!).
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Invert A by row operations on [A I] to reach [I A-I].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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).
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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.