- 57.1: What is the difference between the sum of 12 and 12 and theproduct ...
- 57.2: homas Jefferson was born in 1743. How old was hewhen he was elected...
- 57.3: Subtract 34 from 56. Use 12 as the common denominator.
- 57.4: 12 23
- 57.5: 12 16
- 57.6: 56 23
- 57.7: How many 3535 s are in 34? (34 35 )
- 57.8: $32.50 10
- 57.9: 4 (12 0.2)
- 57.10: 6 0.12
- 57.11: 5 3 38 258
- 57.12: 34 55 3
- 57.13: Fifty percent of this rectangle is shaded.Write 50% as a reduced fr...
- 57.14: What is the place value of the 7 in 3.567?
- 57.15: Divide 0.5 by 4 and round the quotient to the nearesttenth.
- 57.16: Arrange these numbers in order from least to greatest:0.3, 3.0, 0.03
- 57.17: In this sequence the first term is 2, the second term is 4, andthe ...
- 57.18: In a deck of 52 cards there are four aces. What is the ratio of ace...
- 57.19: a. Calculate the perimeter of therectangle shown. b. Multiply the l...
- 57.20: What number is 58 of 80?
- 57.21: List the factors of 29.
- 57.22: What is the least common multiple of 12 and 18?
- 57.23: Compare: 58 6 710
- 57.24: What temperature is shown on thisthermometer?
- 57.25: If the temperature shown on thisthermometer rose to 12F, then how m...
- 57.26: Reduce: 2 2 3 3 5 72 2 5 5 7 7
- 57.27: What fraction of the group of circles isshaded?
- 57.28: Ling has a 9-inch stack of CDs on the shelf. Each CD is in a 38-inc...
- 57.29: Subtract 12 from 45. Use 10 as the common denominator
- 57.30: The diameter of a regulation basketball hoop is 18 inches. What is ...
Solutions for Chapter 57: Adding and Subtracting Fractions: Three Steps
Full solutions for Saxon Math, Course 1 | 1st Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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.)
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
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.
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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Every v in V is orthogonal to every w in W.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Solvable system Ax = b.
The right side b is in the column space of A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.