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Math 152 - Linear Systems Test #1, Version A (T/Th sections) Spring, 2010 University Of British Columbia Name: ---------------------------------------------------- ID Number: ------------------------------------------------- Instructions • You should have six pages including this cover. • There are 2 parts to the test: • • • • • • • • o Part A has 10 short questions worth 1 mark each o Part B has 3 long questions worth 5 marks each Although all questions in each part are worth the same, some may he much more difficult - do the easy questions first! Use this booklet to answer questions. Return this exam with your answers. Please show your work. Correct intermediate steps may earn credit. No calculators are permitted on the test. No notes are permitted on the test. Maximum score= 25 Marks (attempt all questions) Maximum Time= 50 minutes. GOOD LUCK! Part A Total B1 B2 B3 Total 10 5 5 5 25 l\'lath 152 Test #1 version A Part A - Short Answer Questions, 1 mark each For Questions A1-A3 below, let AI: Compute 2a ­ ( ~I A2: Compute II all a (1,-2,3) b (-1,0,1) b .... tt ( \0) - ( - I ) 0 ) I") :: A3: Compute a· b -I +0 +3 -: 2 A4: Rukia and her brother Byakuya each received some money from their parents (possibly different amounts). Then when you see them shortly afterwards, neither of them will teU you how much money they have received . Instead, Rukia tells you how much more money she has than Byakuya. Then Byakuya tells you the total amount of money given to them by the parents. Based on these statements from Rukia and Byakuya, wou Id you be able to determine how much money each of the two have received, and why? Circle the answer with the correct reasoning below: (a) Yes, because the statements represent a system of two equations and two unknowns and therefore has a solution. @ Yes, because the statements represent a system of equations that may be reduced to only one solution. (c) No, because the statements represent contradictory equations. (d) No, because the statements represent a syst.em that contains too many variables and not enough useful equations. A5: Consider the augmented matrix below: 2 3 1 2 1 [ 000 o i1 Decide whether the corresponding system has no solutions, exactly one solution or an infinite number of solutions. Justify briefly. \V\h~tt 0-- l\UV\1W tb ~~~C ~r~~1 Math 152 Test #1 version A A6: Consider the following lines of MATLAB code: A = [1 5 -3; 3 2 1J; A (2., 2) What will the output be if these lines are typed into IVIATLAB? 2­ A 7: Consider the following lines of MATLAB code that modify a previously defined 4 x 5 matrix A for i=1:2 ACi,:) A(i,:)+A(4,:); end Circle the case below that correctly describes the action of this code: (a) Exchanges the first and second rows of A. (b) Multiplies each row of A by its row number. (c) Adds the fourth column of A to the first and second columns of A. @ Adds the fourth row of A to the first and second rows of A. A8: Are the vectors (1,2,3) and (-1,0,1) linearly independent? Justify ~ s. ~~ Q . ~. (~f~ duv-ec1lCf\ltS} (\O-t S~ briefly· I~~L J M V Ih'pi.9s) A9: Find a nonzero vector orthogonal to both (1,1, 1) ~ cl (-2,3,7). 1kt- G I'oS.t p~rl~ ~ () ro'(JQ.f f7J AIO: A lineal system with three equations and all solutions lie on the line j I k r t~ree u~n -= vns is such that x = (2,1,0) + t(l, I, 1) What is the reduced row echelon form of the augmented matrix of this system? I ( 0 -I ' -1 I o \ o C? ott ) ( '+) - ~ ) S- . l'vlath 152 Test # 1 version A Part B - Long Answer Questions, 5 marks each B 1: Consider the linear system for the unknowns x, y and .z: + 2y + 3z X - Y + 2.z x + 4y + 9z X 9 9 25 (a) [1 mark] Write the system in an augmented matrix. (b) [3] Do row operations (Gaussian elimination) on the augmented matrix to change it to upper triangular form. (c) [1] Find the solution to the problem using backward substitution from the form above. Hint: this system has a unique solution and all components of the solution are integers. , CiA) (b) ( '­ 1 -- \ ? t 4~' 2 f\. -~ 0 0 -I ~ 1- 9 9 ( ~ i 1..5 ~ 0 ICo 'l... (l')-(l) C~)-L I) I\. 2>' 0 -~ - I 0 0 0 -~ lCo l!a \ Cc. ) """"3 :: 3 - 3'h i, -2 2> .J. =-0 =) ~ : <3 "'2:- - j . =-) X, : 2w 4 q C~)+~~) Math 152 Test #1 version A B2: The line L passes through the points (1,0, -1) and (0,1,2). (a) [2 marks] Find a parametric equation for L. (b) [1] Show that L is parallel to the plane P with equation 2x - Y + z = -2. (c) [2] Find an equation form of the plane that is parallel to the plane P and contains the line L . ~ ~ L: , L [s fur~MlM' c'6 ~ ( - I, \ ) j) d..J,'H ~ ~ ~ cAiv, Y\..oY~ ('2., - 1, f) ~ :: - 0 , -( ) ~ P. - V\ a. (I, ro P if p[;.,r~ l \.d lS - L'/~/-I)+-t (-I,I/~) = ~ l (\:J) ~ (0 , " L, ) l V\ = -2 -1 +- ~ :: 0 -/ L- ~5 p~!~ t-o p. ~ p~ ~s l(\.~oJ. n o.Jov.e ro &e r~ ~ P . If- tv\v~+- 0.)<;0 ~VL ~ (c ) r 0 tlAl (II 01 $ 0 - \ ) r- 2't- ~~t: ::N~ ~ avv.cJ.- ~ fflI\. l- 1· 5 r ovUo <JV\,hl'·.·;s ~ V ~ oLt·~G4.~SI~ ~ / oJa ~ POf WJ (0 I ( I ~ ~~ l.) I'S (i"V\... ~b r~ Math 152 Test #1 version A B3: A line L is given in equation form as Xl - X2 + X3 = + 5 X2 - 2XI X3 = 2 ( l) 1 ( 1. ) (a) [1 mark] Is the point (1 , 1, 1) on the line? Justify briefly. (b) [2] Find a line direction vector for L . (b) [2] ·Write a linear system for ( X l , X2 , X3) whose unique solution is the point on L tha t is closest to the point (2,1,2). It is not necessary to solve the system you write down . (~ ) ~ "f.. =- C" l" I) ) "\ - XL ~ hVSt- e.~ dvt ~ is Q...4Q.. So \t- {S (AJ>+ mA- k q - - ~ 1- n 1- /\ A... ~ ~ 1W po ,vJ i~ ((,') ( () 2, ('1.-) cv{ru~, 1 - (L,',2) Q, ~ :) - 1- S I \ - 0 \ VV\ \J~ ~ 1 cLv~sl- tu VV\V\t L:£ - l2, IlL))-=- /\ A. 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