 4.4.1: Choose the correct response in Exercises 17.Which expression repres...
 4.4.2: Choose the correct response in Exercises 17.Which expression repres...
 4.4.3: Choose the correct response in Exercises 17.Which expression repres...
 4.4.4: Choose the correct response in Exercises 17.Suppose that Ira Specto...
 4.4.5: Choose the correct response in Exercises 17.According to Natural Hi...
 4.4.6: Choose the correct response in Exercises 17.How far does a car trav...
 4.4.7: Choose the correct response in Exercises 17.What is the speed of a ...
 4.4.8: What is the speed of a plane that travels at a rate of 650 mph agai...
 4.4.9: Exercises 9 and 10 are good warmup problems. Refer to the sixstep...
 4.4.10: Exercises 9 and 10 are good warmup problems. Refer to the sixstep...
 4.4.11: Write a system of equations for each problem, and then solve the sy...
 4.4.12: Write a system of equations for each problem, and then solve the sy...
 4.4.13: Write a system of equations for each problem, and then solve the sy...
 4.4.14: Write a system of equations for each problem, and then solve the sy...
 4.4.15: If x units of a product cost C dollars to manufacture and earn reve...
 4.4.16: If x units of a product cost C dollars to manufacture and earn reve...
 4.4.17: Write a system of equations for each problem, and then solve the sy...
 4.4.18: Write a system of equations for each problem, and then solve the sy...
 4.4.19: Joyce Nemeth bought each of her seven nephews a gift, either a DVD ...
 4.4.20: Jason Williams bought each of his five nieces a gift, either a DVD ...
 4.4.21: Karen Walsh has twice as much money invested at 5% simple annual in...
 4.4.22: Glenmore Wiggan invested some money in two accounts, one paying 3% ...
 4.4.23: Two of the topgrossing North American concert tours in 2008 were T...
 4.4.24: Two other topgrossing North American concert tours in 2008 were Ce...
 4.4.25: Write a system of equations for each problem, and then solve the sy...
 4.4.26: Write a system of equations for each problem, and then solve the sy...
 4.4.27: Write a system of equations for each problem, and then solve the sy...
 4.4.28: Write a system of equations for each problem, and then solve the sy...
 4.4.29: How many pounds of nuts selling for $6 per lb and raisins selling f...
 4.4.30: Jasmine Vazquez, who works at a delicatessen, is preparing a cheese...
 4.4.31: Write a system of equations for each problem, and then solve the sy...
 4.4.32: Write a system of equations for each problem, and then solve the sy...
 4.4.33: Write a system of equations for each problem, and then solve the sy...
 4.4.34: Write a system of equations for each problem, and then solve the sy...
 4.4.35: Write a system of equations for each problem, and then solve the sy...
 4.4.36: Write a system of equations for each problem, and then solve the sy...
 4.4.37: Write a system of equations for each problem, and then solve the sy...
 4.4.38: Write a system of equations for each problem, and then solve the sy...
 4.4.39: Write a system of equations for each problem, and then solve the sy...
 4.4.40: Write a system of equations for each problem, and then solve the sy...
 4.4.41: At the beginning of a bicycle ride for charity, Yady Saldarriaga an...
 4.4.42: Humera Shams left Farmersville in a plane at noon to travel to Exet...
 4.4.43: Graph each linear inequality. See Section 3.5.x + y 4
 4.4.44: Graph each linear inequality. See Section 3.5.y 3x + 2
 4.4.45: Graph each linear inequality. See Section 3.5.3x + 2y 6 0
Solutions for Chapter 4.4: Applications of Linear Systems
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 4.4: Applications of Linear Systems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Chapter 4.4: Applications of Linear Systems includes 45 full stepbystep solutions. Since 45 problems in chapter 4.4: Applications of Linear Systems have been answered, more than 36563 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = 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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
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.

Multiplier eij.
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).

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).