- 1.1: What is the sum of 25 and 40?
- 1.2: At a planetarium show, Johnny counted 137 students and 89 adults.He...
- 1.3: What is the difference when 93 is subtractedfrom 387?
- 1.4: Keisha paid $5 for a movie ticket that cost $3.75. Find how muchcha...
- 1.5: Tatiana had $5.22 and earned $4.15 more by taking care of herneighb...
- 1.6: The soup cost $1.25, the fruit cost $0.70, and the drink cost $0.60...
- 1.7: 6347+ 50
- 1.8: 63257+ 198
- 1.9: 789+ 987
- 1.10: 432579+ 3604
- 1.11: 345 67
- 1.12: 678 416
- 1.13: 3764 96
- 1.14: 875 + 1086 + 980
- 1.15: 10 + 156 + 8 + 27
- 1.16: $3.47 $0.92
- 1.17: $24.15 $1.45
- 1.18: $0.75+ $0.75
- 1.19: $0.12$0.46+ $0.50
- 1.20: What is the name for the answer when we add?
- 1.21: What is the name for the answer when we subtract?
- 1.22: The numbers 5, 6, and 11 are a fact family. Form twoaddition facts ...
- 1.23: Rearrange the numbers in this addition fact to form anotheraddition...
- 1.24: Rearrange the numbers in this subtraction fact to formanother subtr...
- 1.25: Describe a way to check the correctness of a subtraction answer.
Solutions for Chapter 1: Adding Whole Numbers and Money Subtracting Whole Numbers and Money Fact Families, Part 1
Full solutions for Saxon Math, Course 1 | 1st Edition
Solutions for Chapter 1: Adding Whole Numbers and Money Subtracting Whole Numbers and Money Fact Families, Part 1Get Full Solutions
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Invert A by row operations on [A I] to reach [I A-I].
A sequence of steps intended to approach the desired solution.
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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Singular matrix A.
A square matrix that has no inverse: det(A) = 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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.