29 62 CHAPTER 1 Linear Equations in Linear Algebra9. V₁ =10. V₁ =11.13.-[-]---3215.In Exercises 11-14, find the value(s) of h for which the vectorsare linearly dependent. Justify each answer.17.-----=10-3---5-353-57-819. 12-4-5BED AND5-97685h1.[8]096-65439Determine by inspection whether the vectors in Exercises 15-20are linearly independent. Justify each answer.[{][B][B][-]112.14.16.2-9h2HO-47-318.6[][]-29220.4 0₁In Exercises 21 and 22, mark each statement True or False. Justifyeach answer on the basis of a careful reading of the text.h[4][3][³][8]5-7--2530[8]21. a. The columns of a matrix A are linearly independent if theequation Ax = 0 has the trivial solution.b. If S is a linearly dependent set, then each vector is a linearcombination of the other vectors in S.c. The columns of any 4 x 5 matrix are linearly dependent.d. If x and y are linearly independent, and if {x, y, z) islinearly dependent, then z is in Span {x,y}.22. /a. Two vectors are linearly dependent if and only if they lieon a line through the origin.b. If a set contains fewer vectors than there are entries in thevectors, then the set is linearly independent.c. If x and y are linearly independent, and if z is inSpan {x, y), then {x, y, z) is linearly dependent.d. If a set in R" is linearly dependent, then the set containsmore vectors than there are entries in each vector.In Exercises 23-26, describe the possible echelon forms of thematrix. Use the notation of Example 1 in Section 1.2.23. A is a 3 x 3 matrix with linearly independent columns.24. A is a 2 x 2 matrix with linearly dependent columns,25. A is a 4 × 2 matrix, A = [a, a₂], and a2 is not aal.a₂}26. A is a 4 x 3 matrix, A = [a, a2 a3], such that {a,, a2) islinearly independent and a3 is not in Span {a,, a2}.27. How many pivot columns must a 7 x 5 matrix have if itscolumns are linearly independent? Why?28. How many pivot columns must a 5 x 7 matrix have if itscolumns span R5? Why?29. Construct 3 x 2 matrices A and B such that Ax = 0 has onlythe trivial solution and Bx = 0 has a nontrivial solution.30. a. Fill in the blank in the following statement: "If A isan m x n matrix, then the columns of A are linearlyindependent if and only if A has pivot columns."b. Explain why the statement in (a) is true.Exercises 31 and 32 should be solved without performing rowoperations. [Hint: Write Ax = 0 as a vector equation.]31. Given A =multiple of2 3 532. Given A =-5 1-3 -11 0-4-414-7is the sum of the first two columns. Find a nontrivial solutionof Ax = 0.1659 -3observe that the third column3 , observe that the first column3plus twice the second column equals the third column. Finda nontrivial solution of Ax = 0.Each statement in Exercises 33-38 is either true (in all cases)or false (for at least one example). If false, construct a specificexample to show that the statement is not always true. Such anexample is called a counterexample to the statement. If a statementis true, give a justification. (One specific example cannot explainwhy a statement is always true. You will have to do more workhere than in Exercises 21 and 22.)33. If v₁,..., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4)is linearly dependent.34. If V₁,..., V4 are in R4 and v3 = 0, then {V₁, V2, V3, V4) islinearly dependent.35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of V₁.then {V₁, V₂} is linearly independent.36. If V₁,..., V4 are in R4 and v3 is not a linear combination ofV₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent.37. If V₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent.then {V1, V2, V3, V4) is also linearly dependent.38. If V₁,..., V4 are linearly independent vectors in R4, then{V1, V2, V3} is also linearly independent. [Hint: Think aboutXIVI + X₂V₂ + X3V3 +0 V4 = 0.]1.

Solution to the given question, Introduction: The solution of a matrix equation AX=0 is a trivial…

Answered: 29. Construct 3 x 2 matrices A and B…

29. Construct 3 x 2 matrices A and B such that Ax = 0 has only the trivial solution and Bx = 0 has a nontrivial solution.

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 7SA

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(Practice) State whether the following are valid function names and if so, whether they’re mnemonic names that convey some idea of the function’s purpose. If they are invalid names, state why. powerdensity m1234 newamp 1234 abcd total tangent absval computed b34a 34ab volts$ a2B3 while minVal sine $sine cosine speed netdistance sum return stack
(For thought) a. A token of a computer language is any sequence of characters, with no intervening characters or white space, that taken as a unit has a unique meaning. Using this definition of a token, determine whether escape sequences, function names, and the keywords listed in Table 2.1 are tokens of the C++ language. b. Discuss whether adding white space to a message alters the message and whether messages can be considered tokens of C++. c. Using the definition of a token in Exercise 4a, determine whether the following statement is true: “Except for tokens of the language, C++ ignores all white space.”
(Civil eng.) Modify the program written for Exercise 9 to determine the maximum load that can be placed at the end of an 8-foot I-beam, shown in Figure 2.21, so that the stress on the fixed end is 20,000lbs/in2. Use the fact that this beam’s rectangular moment of inertia is 21.4 in4 and the value of c is 3 in.
(Thermodynamics) The work, W, performed by a single piston in an engine can be determined by this formula: W=Fd F is the force provided by the piston in Newtons. d is the distance the piston moves in meters. a. Determine the units of W by calculating the units resulting from the right side of the formula. Check that your answer corresponds to the units for work listed in Table 1.1. b. Determine the work performed by a piston that provides a force of 1000 N over a distance of 15 centimeters.
(Heat transfer) The formula developed in Exercise 5 can be used to determine the cooling time, t, caused only by radiation, of each planet in the solar system. For convenience, this formula is repeated here (see Exercise 5 for a definition of each symbol): t=Nk2eAT3fin A=surfaceareaofasphere=4r2 N=numberofatoms=volumeofthespherevolumeofanatom Volume of a sphere sphere=43radius3 The volume of a single atom is approximately 11029m3 . Using this information and the current temperatures and radii listed in the following chart, determine the time it took each planet to cool to its current temperature, caused only by radiation.
(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.
(Chemistry) a. Determine the final units of the following expression, which provides the molecular weight of 1.5 moles of hydrogen peroxide: 1.5moles34.0146grams/mole b. Determine the final units of the following expression, which provides the molecular weight of 5.3 moles of water: 5.3moles18grams/mole
(Civil eng.) The maximum load that can be placed at the end of a symmetrical wooden beam, such as the rectangular beam shown in Figure 2.20, can be calculated as the following: L=S1dc L is the maximum weight in lbs of the load placed on the beam. S is the stress in lbs/in2. I is the beam’s rectangular moment of inertia in units of in4. d is the distance in inches that the load is placed from the fixed end of the beam (the “moment arm”). c is one-half the height in inches of the symmetrical beam. For a 2” × 4” wooden beam, the rectangular moment of inertia is given by this formula: I=baseheight3=12=24312=10.674 c=(4in)=2in a. Using this information, design, write, compile, and run a C++ program that computes the maximum load in lbs that can be placed at the end of an 8-foot 24 wooden beam so that the stress on the fixed end is 3000lb/in2. b. Use the program developed in Exercise 9a to determine the maximum load in lbs that can be placed at the end of a 3” × 6” wooden beam so that the stress on the fixed end is 3000lb/in2.