- 12.1.1: State whether the events are independent or dependent.
- 12.1.2: State whether the events are independent or dependent.
- 12.1.3: An ice cream shop offers a choice of two types of cones and 15 flav...
- 12.1.4: A bookshelf holds 4 different biographies and 5 different mystery n...
- 12.1.5: Lances math quiz has eight true-false questions. How many different...
- 12.1.6: Pizza House offers three different crusts, four sizes, and eight to...
- 12.1.7: For a college application, Macawi must select one of five topics on...
- 12.1.8: State whether the events are independent or dependent.
- 12.1.9: State whether the events are independent or dependent.
- 12.1.10: Each of six people guess the total number of points scored in a bas...
- 12.1.11: The letters A through Z are written on pieces of paper and placed i...
- 12.1.12: Tim wants to buy one of three different books he sees in a book sto...
- 12.1.13: A video store has 8 new releases this week. Each is available on vi...
- 12.1.14: Carlos has homework in math, chemistry, and English. How many ways ...
- 12.1.15: The menu for a banquet has a choice of 2 types of salad, 5 main cou...
- 12.1.16: A baseball glove manufacturer makes gloves in 4 different sizes, 3 ...
- 12.1.17: Each question on a five-question multiple-choice quiz has answer ch...
- 12.1.18: Abby is registering at a Web site. She must select a password conta...
- 12.1.19: How many ways can you arrange the science books?
- 12.1.20: Since the science books are to be together, they can be treated lik...
- 12.1.21: How many area codes were possible before 1995?
- 12.1.22: In 1995, the restriction on the middle digit was removed, allowing ...
- 12.1.23: How many ways can six different books be arranged on a shelf if one...
- 12.1.24: In how many orders can eight actors be listed in the opening credit...
- 12.1.25: How many different 5-digit codes are possible using the keypad show...
- 12.1.26: Use the Internet or other resource to find the configuration of let...
- 12.1.27: Describe a situation in which you can use the Fundamental Counting ...
- 12.1.28: Explain how choosing to buy a car or a pickup truck and then select...
- 12.1.29: The members of the Math Club need to elect a president and a vice p...
- 12.1.30: Use the information on page 684 to explain how you can count the ma...
- 12.1.31: How many numbers between 100 and 999, inclusive, have 7 in the tens...
- 12.1.32: A coin is tossed four times. How many possible sequences of heads o...
- 12.1.33: Prove that 4 + 7 + 10 + + (3n + 1) = _ n(3n + 5) 2 for all positive...
- 12.1.34: Find the indicated term of each expansion.
- 12.1.35: Find the indicated term of each expansion.
- 12.1.36: Edison is located at (9, 3) in the coordinate system on a road map....
- 12.1.37: Solve each equation by factoring
- 12.1.38: Solve each equation by factoring
- 12.1.39: Solve each equation by factoring
- 12.1.40: Solve each matrix equation
- 12.1.41: Solve each matrix equation
- 12.1.42: Evaluate each expression.
- 12.1.43: Evaluate each expression.
- 12.1.44: Evaluate each expression.
- 12.1.45: Evaluate each expression.
- 12.1.46: Evaluate each expression.
- 12.1.47: Evaluate each expression.
- 12.1.48: Evaluate each expression.
- 12.1.49: Evaluate each expression.
Solutions for Chapter 12.1: The Counting Principle
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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].
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).