 2.7.1: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.2: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.3: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.4: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.5: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.6: Answer each question. See Example 1 and the ProblemSolving Hint pr...
 2.7.7: Solve each percent problem. Remember that base rate percentage.The ...
 2.7.8: Solve each percent problem. Remember that base rate percentage.The ...
 2.7.9: Solve each percent problem. Remember that base rate percentage.The ...
 2.7.10: Solve each percent problem. Remember that base rate percentage.An a...
 2.7.11: Suppose that a chemist is mixing two acid solutions, one of 20% con...
 2.7.12: Suppose that pure alcohol is added to a 24% alcohol mixture. Which ...
 2.7.13: Work each mixture problem. See Example 2.How many liters of 25% aci...
 2.7.14: Work each mixture problem. See Example 2.How many gallons of 50% an...
 2.7.15: Work each mixture problem. See Example 2.A pharmacist has 20 L of a...
 2.7.16: Work each mixture problem. See Example 2.A certain metal is 20% tin...
 2.7.17: In a chemistry class, 12 L of a 12% alcohol solution must be mixed ...
 2.7.18: How many liters of a 10% alcohol solution must be mixed with 40 L o...
 2.7.19: Minoxidil is a drug that has recently proven to be effective in tre...
 2.7.20: A pharmacist wishes to mix a solution that is 2% minoxidil. She has...
 2.7.21: How many liters of a 60% acid solution must be mixed with a 75% aci...
 2.7.22: How many gallons of a 12% indicator solution must be mixed with a 2...
 2.7.23: Work each investment problem using simple interest. See Example 3.A...
 2.7.24: Work each investment problem using simple interest. See Example 3.M...
 2.7.25: Work each investment problem using simple interest. See Example 3.A...
 2.7.26: Work each investment problem using simple interest. See Example 3.W...
 2.7.27: Work each problem involving monetary values. See Example 4.A coin c...
 2.7.28: Work each problem involving monetary values. See Example 4.A bank t...
 2.7.29: Work each problem involving monetary values. See Example 4.In May 2...
 2.7.30: Work each problem involving monetary values. See Example 4.A movie ...
 2.7.31: Work each problem involving monetary values. See Example 4.Harriet ...
 2.7.32: Work each problem involving monetary values. See Example 4.Harriets...
 2.7.33: Solve each problem involving distance, rate, and time. See Example ...
 2.7.34: Solve each problem involving distance, rate, and time. See Example ...
 2.7.35: Solve each problem involving distance, rate, and time. See Example ...
 2.7.36: Solve each problem involving distance, rate, and time. See Example ...
 2.7.37: Solve each problem involving distance, rate, and time. See Example ...
 2.7.38: Solve each problem involving distance, rate, and time. See Example ...
 2.7.39: In Exercises 3942, find the rate on the basis of the information pr...
 2.7.40: In Exercises 3942, find the rate on the basis of the information pr...
 2.7.41: In Exercises 3942, find the rate on the basis of the information pr...
 2.7.42: In Exercises 3942, find the rate on the basis of the information pr...
 2.7.43: Solve each motion problem. See Examples 6 and 7.Atlanta and Cincinn...
 2.7.44: Solve each motion problem. See Examples 6 and 7.St. Louis and Portl...
 2.7.45: Solve each motion problem. See Examples 6 and 7.A train leaves Kans...
 2.7.46: Solve each motion problem. See Examples 6 and 7.Two steamers leave ...
 2.7.47: Solve each motion problem. See Examples 6 and 7.From a point on a s...
 2.7.48: Solve each motion problem. See Examples 6 and 7.At a given hour, tw...
 2.7.49: Solve each motion problem. See Examples 6 and 7.Two planes leave an...
 2.7.50: Solve each motion problem. See Examples 6 and 7.Two trains leave a ...
 2.7.51: Solve each motion problem. See Examples 6 and 7.Two cars start from...
 2.7.52: Solve each motion problem. See Examples 6 and 7.Two cars leave town...
 2.7.53: Solve each problem.Kevin is three times as old as Bob. Three years ...
 2.7.54: Solve each problem.A store has 39 qt of milk, some in pint cartons ...
 2.7.55: Solve each problem.A table is three times as long as it is wide. If...
 2.7.56: Solve each problem.Elena works for $6 an hour. A total of 25% of he...
 2.7.57: Solve each problem.Paula received a paycheck for $585 for her weekl...
 2.7.58: Solve each problem.At the end of a day, the owner of a gift shop ha...
 2.7.59: Decide whether each statement is true or false. See Section 1.4.6 7 6
 2.7.60: Decide whether each statement is true or false. See Section 1.4.10 10
 2.7.61: Decide whether each statement is true or false. See Section 1.4.4 3
 2.7.62: Decide whether each statement is true or false. See Section 1.4.11...
 2.7.63: Decide whether each statement is true or false. See Section 1.4.0 7...
 2.7.64: Graph the numbers on a number line. 3,  See Section 1.4.
Solutions for Chapter 2.7: Further Applications of Linear Equations
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 2.7: Further Applications of Linear Equations
Get Full SolutionsBeginning Algebra was written by and is associated to the ISBN: 9780321673480. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.7: Further Applications of Linear Equations includes 64 full stepbystep solutions. Since 64 problems in chapter 2.7: Further Applications of Linear Equations have been answered, more than 37848 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
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 nl  An).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Elimination.
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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.