- 14.1.1: The ________ of A and B consists of all elements in either A or B o...
- 14.1.2: The ________ of A with B consists of all elements in both A and B.
- 14.1.3: True or False The intersection of two sets is always a subset of th...
- 14.1.4: True or False If A is a set, the complement of A is the set of all ...
- 14.1.5: If each element of a set A is also an element of a set B, we say th...
- 14.1.6: If the number of elements in a set is a nonnegative integer, we say...
- 14.1.7: If A and B are finite sets, the Counting Formula states that n(A B) =
- 14.1.8: True or False If a task consists of a sequence of three choices in ...
- 14.1.9: Write down all the subsets of 5a, b, c, d6.
- 14.1.10: Write down all the subsets of 5a, b, c, d, e6.
- 14.1.11: If and find n1A B2. n1A2 = 15, n1B2 = 20, n1A B2 = 10,
- 14.1.12: If and find n1A B2. n1A2 = 30, n1B2 = 40, n1A B2 = 45
- 14.1.13: If and find n1A2. n1A B2 = 50, n1A B2 = 10, n1B2 = 20
- 14.1.14: If and find n1A2. n1A B2 = 60, n1A B2 = 40, n1A2 = n1B
- 14.1.15: In 1522, use the information given in the figure
- 14.1.16: In 1522, use the information given in the figure
- 14.1.17: In 1522, use the information given in the figure
- 14.1.18: In 1522, use the information given in the figure
- 14.1.19: In 1522, use the information given in the figure
- 14.1.20: In 1522, use the information given in the figure
- 14.1.21: In 1522, use the information given in the figure
- 14.1.22: In 1522, use the information given in the figure
- 14.1.23: Shirts and Ties A man has 5 shirts and 3 ties. How many different s...
- 14.1.24: Blouses and Skirts A woman has 5 blouses and 8 skirts. How many dif...
- 14.1.25: Four-digit Numbers How many four-digit numbers can be formed using ...
- 14.1.26: Five-digit Numbers How many five-digit numbers can be formed using ...
- 14.1.27: Analyzing Survey Data In a consumersurvey of 500 people, 200 indica...
- 14.1.28: Analyzing Survey Data In a student survey, 200 indicated that they ...
- 14.1.29: Analyzing Survey Data In a survey of 100 investors in the stock mar...
- 14.1.30: Classifying Blood Types Human blood is classified as either Blood i...
- 14.1.31: Demographics The following data represent the marital status of mal...
- 14.1.32: Demographics The following data represent the marital status of fem...
- 14.1.33: Stock Portfolios As a financial planner, you are asked to select on...
- 14.1.34: Make up a problem different from any found in the text that require...
- 14.1.35: Investigate the notion of counting as it relates to infinite sets.W...
Solutions for Chapter 14.1: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry | 9th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Upper triangular systems are solved in reverse order Xn to Xl.
Column space C (A) =
space of all combinations of the columns of A.
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.
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
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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).
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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 ().
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.