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# LinearEngineeringSystems ENGR231

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This 11 page Class Notes was uploaded by Lyda Ryan I on Wednesday September 23, 2015. The Class Notes belongs to ENGR231 at Drexel University taught by OlehTretiak in Fall. Since its upload, it has received 83 views. For similar materials see /class/212338/engr231-drexel-university in Engineering and Tech at Drexel University.

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Date Created: 09/23/15

ENGR 231 Linear Engineering Spring 2011 Practice Problems for Exam I Solution 1 Find the general solution of the following system of equations Write your answer in parametric vector form x1 5x2 9x3 8164 7 x2 3xS 4x4 2 2x2 6x3 8164 4 Answer Augmented matrix and RREF 1 5 9 8 7 0 1 3 4 2 0 2 6 8 4 12 3 0 0 OOH Ot O OWGN N Basic variables 1 2 Free variables 3 4 x16x3 12c4 3 x1 3 6x3 12x4 3 6 12 x2 3x3 4x4 2 x2 2 3x3 4x4 2 3 4 x3 x4 x3 free x3 x3 0 1 0 x4 free x4 x4 0 0 1 2 Find the product of the matrixA and the vector x 2 l 4 12 A 1 3 3 8 Answer 4 12 2 4 12 8 12 20 1 3 12 1 1 3 23 5 3 8 3 8 6 8 14 3 Are the following matrices in echelon form Are they in the reduced echelon form Justify your answer 1234 1 804 0282 001 2 a b 0017 0000 0000 0000 Answer a Echelon form values of 0 below and to the left of each pivot b Reduced echelon form 1 s on pivot positions 0 above and below each pivot ENGR 231 Spring 2011 Exam 1 Practice Problems Solutions 4 The arrays in Problem 3 are augmented matrices for linear systems For each matrix answer the following questions a Identify the free and basic variables b State whether the systems are consistent c For consistent systems nd a particular solution d For systems with a free variable nd a nontrivial solution to the homogeneous equation e Verify that the sum of a particular and homogenous solutions is also a solution of the linear system 1 Write the solution set in parametric vector form Answer For matrix 3 a 1 2 and 3 are basic there are no free variables The system is consistent To go further we convert the matrix to reduced echelon form getting 1 0 0 37 0 1 0 27 0 0 1 7 0 0 0 0 The particular solution is x1 37 x2 727 x3 7 The only solution to the homogeneous system is x 0 For verification 1 2 3 37 37 54 21 4 0 2 8 54 56 2 27 0 0 1 7 7 7 0 0 0 0 0 Nere is no need to check the homogenous system because 0 is always a solution The solution in paramteric vector form is For matrix 3 b 1 and 3 are basic 2 is free The system is consistent x1 4 8x2 4 8 x2 x2 0 X2 1 xi 2 2 0 4 8 P 0 V 1 2 0 ENGR 231 Spring 2011 Exam 1 Practice Problems Solutions 2 1 8 0 4 12 0 0 1 2 Xpv 1 A b 2 0 0 0 0 0 0 0 0 1 8 0 12 4 12 8 4 0 AXb 0 0 1 1 2 2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 5 Vectors a1 a2 and b are de ned below For what value or values of h is b a linear combination of a1 and a2 1 3 h a1 a2 1 b 5 2 2 3 Answer 1 3 h 1 3 h 1 3 h 0 1 5 0 1 5 N 0 1 5 2 2 3 0 8 3 2h 0 0 43 2h For consistency h215 To check we complete the row reduction process and get 1 0 h 15 0 1 5 0 0 43 2h If h is a linear combination then 61211 62211 b We set h 215 CI 141565 oz 5 and get 1 3 215 65 0 5 1 5 a 2 2 3 which veri es the answer ENGR 231 Spring 2011 Exam 1 Practice Problems Solutions 6 A company manufactures two products For each unit of product 1 the company spends 4 on materials 8 on labor 1 on packaging and 5 on overhead expenses For each unit of product 2 the company spends 6 on materials 10 on labor 2 on packaging and 5 on overhead The company wants to know how much of each product to make in order to use exactly all of its budgeted resources of 600 for materials 1100 for labor 175 for packaging and 600 for overhead a Set up but do not solve a vector equation that describes this problem Include a statement about what the variables in the equation represent b Write an equivalent matrix equation for this problem Do not solve it Answer The vector represents the cost of a unit of the product The first element is the cost of materials the second the cost of labor third the cost of packaging and fourth the overhead Therefore the unit cost vector for product 1 is V1 4 8 1 5 and for product 2 it is V2 6 10 2 5 Let x1 be the amount of product 1 and x the amount of product 2 The equation to be satisfied is 4 6 600 XI 8 x2 10 1100 1 2 175 5 5 600 4 6 600 8 10 Xi 1100 1 2 x2 175 5 5 600 7 Mark the following statements as True or False Justify your answer a Two matrices are row equivalent if they have the same number of rows Answer False We can have matrices such as a nonzero and zero matrices that have the same number of rows but are not row equivalent b A nonzero matrix has a unique echelon form Answer False The reduced echelon form is unique but not the ordinary echelon form The pivor locations however are unique and present in both echelon forms 8 Boron sulfide reacts violently with water to form boric acid and hydrogen sulfide gas The unbalanced equation is BZS3 H20 a H3BO3 H28 For each compound construct a vector describing the component atoms Write a vector equation with unknowns that must be found to balance the equation Solve the system and balance the chemical equation Answer B2S3 N H20 N H3B03 N HzS N DOWN HGOO Uon OleO ENGR 231 Spring 2011 Exam 1 Practice Problems Solutions me a 6 Z 3 15 B253 sto a ZH3B03 3st the tram m cars per mmute m Answer39The reduced echelon form 15 1 0 1 0 20 0 1 71 0 60 0 From 0 0 1 60 the reduced echelon form x 20 ex Smeex 2 o the largestvalue ofxgxs 20 and occurs when x o ENGR Z31 Spnng 2011 Exam 1 Pracuce Problems Soluuons 10 Mark each of the following statements as True or False Justify your answer aIn some cases a matrix may be row reduced to more than one matrix in reduced echelon form using different sequence of row operations Answer False The reduced echelon form is unique bThe row reduction algorithm applies only to augmented matrices for a linear system Answer False Row reduction may be applied to any matrix Useful information can be obtained if we apply it to the coefficient matrix For example we can nd out if there are free variables which will tell us if the solution is unique c No matter what value of A there is always some value of b for which the equation Ax b is consitent Answer True b 0 always produces a consistent system d If x is a nontrivial solution of Ax 0 then every entry in x is nonzero Anwer False Let A 0 0 0 1 x2 1 0 is a nontirivial solution with a zero entry 10 A system of linear equations with more equations than unknowns is sometimes called an overdetermined system Can such a system be consistent Illustrate your answer with a specific system of three equations in two unknowns Answer An overdetermined system can be consistent For example if the system is homogeneous it is always consistent 1 2 3 3 4 K 7 5 5 10 is consistent The solution is x 1 1 11 If A is a 2x4 matrix with two pivot positions does the equation Ax 0 have a nontrivial solution Does the equation Ax b have at least one solution for every possible b Answer A has 2 basic variables and 2 free variables Therefore the homogeneous equation has nontrivial solutions The since there is a pivot in each row the equationAx b has a solution for all h ENGR 231 Spring 2011 Exam 1 Practice Problems Solutions 6 EN GR 231Linear Engineering Systems Mock Final Fall 2009 1 Important The nal exam will be comprehensive in coverage Review the textbook sections covered in the homework assign ments In ao lo lition7 MATLAB basics based on the laboratory manuals will be covered on the exam EN GR 231Linear Engineering Systems Mock Final Fall 2009 2 11 to 9 g 71 9 to 1 Let A be an n x 71 matrix Find the general solution of the following system of equations Write your answer in parametric vector orm 11 7 512 7 913 814 77 127173137414 2 212 613 7 814 4 1 For each matrix below determine whether its columns form a linearly independent set Also determine whether the columns of the matrix span 9931 Give reasons for your answers 74 12 2 7 0 1 5 73 2 1 73 74 76 5 0 4 79 18 73 8 6 13 73 0 0 0 0 Use the inverse of a Matrix to solve the following system 511 7 612 1 7 711 812 73 Write statements from the 1nvertible Matrix Theorem that are each equivalent to the statement 77A is invertible77 Use the following concepts one in each statement a Nul A b det A c basis In each case either explain why H is a subspace of R 3 or explain why H is not a subspace I 2175yi72 731F271 aH y 241321 bH 33715 st1nR 2 s75t a CH b 3a72b5c c 1 Let H be the set below1 Determine whether H is a subspace of 9931 2a 7 4b 7 Sc H 4a76b abcin9 9a 7 b 70 3a 7b 7 c 1 Let H isggfig 2d a bcde any real numbers 512 7 Sc 7 d 46 a Explain why H is a vector space a subspace of R 41 b Find a set of vectors that spans H1 C Find a basis for H1 The matrices A and B below are row equivalent1 2 74 71 73 5 2 2 74 71 73 5 2 A7 71 2 74 73 77 77 B7 0 0 3 71 3 4 7 3 76 6 77 15 13 7 7 0 0 0 0 0 5 5 710 5 710 20 10 0 0 0 0 0 0 81 Find a basis for the column space of A b What is the dimension of the null space of A1 1 Assume that A a1 a2 a5 and B b1 b2 b5 are row equivalent where 1 2 72 0 7 1 0 4 0 73 72 73 1 71 75 0 1 73 0 5 A 73 74 072 73 7 B 00 0174 3 6 76 5 1 0 0 0 0 0 EN GR 231Linear Engineering Systems Mock Final Fall 2009 H O H H H to H CA3 H H HH H H 50 to O lLetV1 iLetA ai Find a basis for the column space of A b Fill in the blanks NulA is a dimensional subspace of R k7 where k 7 0 2 71 iLetx67u 17V 27andWSpanu7ViNotethatuV0l 4 1 4 Find a vector a in W and a vector b that is orthogonal to W7 such that x a b 3 0 8 i Let u 2 7 V 1 J 7 y 77 J 7 and let W Spanu7 Vi Clearly7 u7V is an orthogonal 1 72 4 basis for W1 Use this fact to nd the point in W closest to y Then compute the distance from y to the subspace W1 1 7 y 2 3 21 w 5 a Show that V1 and V2 is an orthogonal basis for W Span v17 1 72 1 71 1 b Write y as the sum of a vector in W and a vector orthogonal to W 2 72 1 lLetvl 1 7V2 2 7V3 2 2 1 72 a Show that the set VV1 V2 V3 is an orthogonal setl b Show that VV1 V2 V3 is a basis for 993 h 26 a2g b2h c2i 7B dSg e3h f3i 7C 9 h i i 2f c D mw a c g d f 2d 9 i a Suppose that det A5i Find det B7det C7 det AC 1 Calculate the area of the parallelogram determined by the points 2 27 0737 47 17 and 674 Hint Translate the parallelogram to one having the origin as a vertex Find a formula for the area of a the triangle whose vertices are 07 V17 Vgi Solve the equation 14XB 1 C D for X7 assuming A and B are invertible A company manufactures two products For each unit of product 17 the company spends 4 on materials7 8 on labor7 1 on packaging7 and 5 on overhead expenses For each unit of product 27 the company spends 6 on materials7 10 on labor7 2 on packaging7 and 5 on overhead The company wants to know how much of each product to make in order to use exactly all of its budgeted resources of 600 for materials7 1100 for labor7 175 for packaging7 and 600 for overheadl a Set up but do not solve a vector equation that describes this problemi Include a statement about what the variables in the equation representi b Write an equivalent matrix equation for this problemi Do not solve it Balance the following chemical equation using vector equation approach discussed in this section a Boron sul de reacts violently with water to form boric acid and hydrogen sul de gas The unbal anced equation is give by B2S3H20 HHgBOngHQS For each compound7 construct a vector that lists the numbers of atoms of boron7 sulfur7 hydrogen and oxygen lLetb17b21g7x17andb17b2i a Explain how you can tell easily that b17b2 is a basis for R 2 Row operations are unnecessary Your explanation should show that you know the de nition of a basis EN GR 231Linear Engineering Systems Mock Final Fall 2009 4 b Write an equation that connects the vector x and its coordinate vector xlg C Find Mg 21 The set E 1 t1 t27 t 7 t2 is a basis for P2 Find the coordinate vector of pt 6 3t 7 t2 to to to CA3 N F to 01 to on N 77 28 1 Leastsquares Let A 2 5 3 4 relative to B The rst four Laguerre polynomials are 11 7 t7 2 7 4t 7 t27 and 6 7 18t 9t2 7 t3 Show that these polynomials form a basis of 3 Mark each statement either True or False You do not have to justify your answer a In some cases7 it is possible for six vectors to span R 5 b If a matrix A is m X n and if the equation Ax b has a solution for some b7 then the columns of A span R m C If a system of linear equations has two different solutions7 then it has in nitely many solutions 1 Every matrix is row equivalent to a unique matrix in echelon form e If V1 and V2 span a plane in R 3 and if V3 is not in that plane7 then V1V2V3 is a linearly independent set Computer Graphics Write the standard matrix of the transformation T R 2 7 R 2 that reflects vectors through the 11 axis and then rotates them clockwise through 7r2 radians Computer Graphics a Why are homogeneous coordinates for 2Dobjects useful b Why are graphics objects often described by wireframe images777with vertices connected by straight lines C Find the matrix A of the linear transformation T R 2 7gt R 2 that maps the standard basis vectors7 5 1 el and eg onto 1 and 2 respect1vely 1 Using homogeneous coordinates for R 2 nd the 3 X 3 matrix that produces the composite 2D transformation that rst translates points in R 2 by the vector and then performs the 2 73 transformation whose matrix A you found in part b 10 Find the leastsquares solution of Ax b 3 Least squares a Find the equation y 60 611 of the leastsquares line for the data 71 07 07 17 17 07 21 Use the method developed in class b Suppose an experiment produces Ly data 27 57 37 67 47 87 57 107 and a scientist wants to model that data with an equation of the form y B11 7 212 ge m Write the design matrix7 the unknown parameter vector and the observation vector for this problem with the entries lled in Matlab Problems a 10 points Write the MATLAB command used to create a new vector b4 whose elements are the sum of cubes of the elements of b17 b2 and b3 4 15 5 b 10 points For the vectors b1 1 b2 2 b3 12 2 27 7 4 1 2 write the MATLAB command used to create the matrix A 15 2 27 using the vectors 5 12 7 b1 b2 and b3 EN GR 231Linear Engineering Systems Mock Final Fall 2009 c Consider the following code in MATLAB for n010 xn1 sinpin10 end i Describe the functionality of the above piece of code

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