- 42.1: Write 12 and 23 as fractions with denominators of 6. Thenadd the re...
- 42.2: According to some estimates, our own galaxy, the Milky Way, contain...
- 42.3: The rectangular school yard is 120 yards long and 40 yardswide. How...
- 42.4: What number is 40% of 30?
- 42.5: In problems 5 and 6, multiply 12 by a fraction equal to 1 to comple...
- 42.6: In problems 5 and 6, multiply 12 by a fraction equal to 1 to comple...
- 42.7: 4.32 0.6 281
- 42.8: 6.3 0.54
- 42.9: (0.15)2
- 42.10: What is the reciprocal of 67?
- 42.11: Which digit in 12,345 has the same place value as the 6 in 67.89?
- 42.12: What is the least common multiple of 3, 4, and 6?
- 42.13: 535 445
- 42.14: 236 423
- 42.15: 83 12
- 42.16: 65 3
- 42.17: 1 14
- 42.18: 1010 55
- 42.19: Form three different fractions that are equal to 13 . (Hint: Multip...
- 42.20: The prime numbers that multiply to form 35 are 5 and 7.Which prime ...
- 42.21: In three games Almas scores were 12,143; 9870; and 14,261.Describe ...
- 42.22: Estimate the quotient of 817641 . Describe how you performed theest...
- 42.23: How many eggs are in 23 of a dozen? Draw a diagram toillustrate the...
- 42.24: Write 34 with a denominator of 8. Subtract the renamed fractionfrom...
- 42.25: What is the perimeter of this rectangle?
- 42.26: What is the area of this rectangle?
- 42.27: The regular price r minus the discount d equals the saleprice sr d ...
- 42.28: Below we show the same division problem written three different way...
- 42.29: What time is 212 hours after 11:45 a.m.?
- 42.30: a. How many 56 s are in 1?b. Use the answer to part a to find the n...
Solutions for Chapter 42: Renaming Fractions by Multiplying by 1
Full solutions for Saxon Math, Course 1 | 1st Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
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).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.