- 25.1: What is the difference between the sum of 12 and 12 and the sumof 1...
- 25.2: In three tries Carlos punted the football 35 yards, 30 yards, and37...
- 25.3: Earths average distance from the Sun is one hundred forty-nine mill...
- 25.4: What is the perimeter of the rectangle?
- 25.5: A 30-inch length of ribbon was cut into 4 equal lengths. Howlong wa...
- 25.6: Two thirds of the class finished the test on time. What fraction of...
- 25.7: Compare: 12 of 12 13 of 12
- 25.8: What fraction is half of the fraction that is half of 12?
- 25.9: A whole circle is 100% of a circle. What percent of a circle is 19 ...
- 25.10: How many 16 s are in 1? How many 16 s are in 12?
- 25.11: What fraction of a circle is 33 13% of a circle?
- 25.12: Divide 365 by 7 and write the answer as a mixed number.
- 25.13: 23 23
- 25.14: 66 56
- 25.15: 30 40 60
- 25.16: 512 512
- 25.17: A team won seven of the twenty games played and lost the rest. What...
- 25.18: Cheryl bought 10 pens for 25 each. How much did she payfor all 10 p...
- 25.19: What is the greatest common factor (GCF) of 24 and 30?
- 25.20: What number is 1100 of 100?
- 25.21: Find each unknown number. Check your work.58 m 1
- 25.22: Find each unknown number. Check your work.144n 12
- 25.23: What is the sum of 3142, 6328, and 4743 to the nearestthousand?
- 25.24: Two thirds of the 60 students liked peaches. How manyof the student...
- 25.25: Estimate the length in inches of the line segment below. Thenuse an...
- 25.26: To divide a circle into thirds, Jan imagined the circle was theface...
- 25.27: Write154 as a mixed number.
- 25.28: Draw and shade rectangles to illustrate and complete thiscomparison...
- 25.29: What are the first four multiples of 25?
- 25.30: Which of these numbers is divisible by both 9 and 10?How do you kno...
Solutions for Chapter 25: Writing Division Answers as Mixed Numbers Multiples
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.)
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).
Column space C (A) =
space of all combinations of the columns of 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).
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.
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].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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.
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).
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.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.