- 2.1: If the factors are 7 and 11, what is the product?
- 2.2: What is the difference between 97 and 79?
- 2.3: If the addends are 170 and 130, what is the sum?
- 2.4: If 36 is the dividend and 4 is the divisor, what is the quotient?
- 2.5: Find the sum of 386, 98, and 1734.
- 2.6: Fatima spent $2.25 for a book. She paid for it with a five-dollar b...
- 2.7: Luke wants to buy a $70.00 radio for his car. He has $47.50. Findho...
- 2.8: Each energy bar costs 75. Find the cost of one dozen energybars. Ex...
- 2.9: 312 86
- 2.10: 4106+ 1398
- 2.11: 4000 1357
- 2.12: $10.00 $2.83
- 2.13: 405(8)
- 2.14: 25 25
- 2.15: 2886
- 2.16: 22515
- 2.17: $1.25 8
- 2.18: 400 50
- 2.19: 1000 8
- 2.20: $45.00 20
- 2.21: Use the numbers 6, 8, and 48 to form two multiplicationfacts and tw...
- 2.22: Rearrange the numbers in this division fact to form anotherdivision...
- 2.23: Rearrange the numbers in this addition fact to form anotheraddition...
- 2.24: a. Find the sum of 9 and 6. b. Find the difference between 9 and 6.
- 2.25: The divisor, dividend, and quotient are in these positions when we ...
- 2.26: Multiply to find the answer to this addition problem:39 + 39 + 39 +...
- 2.27: 365 0
- 2.28: 0 50
- 2.29: 365 365
- 2.30: How can you check the correctness of a division answer thathas no r...
Solutions for Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2
Full solutions for Saxon Math, Course 1 | 1st Edition
Solutions for Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2Get Full Solutions
A = CTC = (L.J]))(L.J]))T for positive definite 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).
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.
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].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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 .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
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.