Math 2360 Exam II 23 July 2008Form B - Make UpFor each of the following problems at least one of the choices is correct. For many of the problems more thanone of the choices is correct. Record your answers for each problem on the answer sheet (last page). Turn in(only) the answer sheet (with your answers on it) at the end of the exam. Do your own work. You may keepthe exam for your own records.1. (12 pts) Consider the following four systems of linear equations:i. ii. 1212 312 302223 1xxxx xxx x−=⎧⎪−+ =−⎨⎪−+ + =⎩12 312 313322151xx xxxxxx−+ =⎧⎪−+=−⎨⎪−−=⎩iii. iv.1232312 34123xxxxxxx x−+ =⎧⎪−+ =−⎨⎪−− =⎩12312312 3132221xxxxxxxx x−+ =⎧⎪−+=−⎨⎪−+ =⎩Which of the above fours systems of linear equations can be solved by Cramer’s Rule?2. (12 pts) Consider the set . Define “addition” by12 12{( , ) | , }TVxxxx=∈\. (We will use symbol for “addition” since this is not12 12 1 122(, ) (, ) ( , )TT Txxyyxyxy⊕=+⊕the usual addition for order pairs.) Define scalar multiplication by . 12 1 2(, ) ( , )xxxxααα=(This is usual scalar multiplication for ordered pairs.)Recall the vector space axioms for addition for a vector space V. i. Addition is commutative, i.e. for every x, y in V we have ⊕=⊕xyyxii. Addition is associative, i.e., for every x, y, z in V we have () ()⊕⊕=⊕ ⊕xy zx yziii. Existence of an additive identity in V, i.e. there exists an additive identity (called 0) in V so thatfor every x in V we have ⊕=0xxiv. Existence of additive inverses in V, i.e. for each x in V there exists an additive inverse (called -x)in V so that ⊕− =0xxWhich of the above axioms are not satisfied by V with its defined “addition” and scalar multiplication?3. (9 pts) Consider the following subsets of :2\i. ii.12 1 2{( , ) | 3 2 0}Txx x x−=12 1 2{( , ) | 1}Txx x x−=−iii. 2212 1 2{( , ) | 0}Txx x x+=Which of the above subsets of are subspaces of ?2\2\4. (9 pts) Consider the following subsets of :3Pi. ii. 3{|(1)(1)1}pPp p∈−−=3{ | () () }pPpx px x′∈−=iii. 3{|(0)1}pPp′′∈=Which of the above subsets of are subspaces of ?3P3P5. (12 pts) Consider the following subsets of :3\i. ii.1211, 1 , 121 2⎧⎫−−⎡⎤⎡⎤⎡⎤⎪⎪⎢⎥⎢⎥⎢⎥−⎨⎬⎢⎥⎢⎥⎢⎥⎪⎪⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⎩⎭1221, 1 , 1121⎧⎫−−⎡⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥−⎣⎦⎣ ⎦⎣ ⎦⎩⎭iii. iv. 21102, 0 , 2, 2112 1⎧⎫−⎡⎤⎡ ⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥−−⎣⎦⎣ ⎦⎣ ⎦⎣ ⎦⎩⎭11121, 2 , 1, 11221⎧⎫−−⎡⎤⎡ ⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥−−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎣⎦⎣ ⎦⎣ ⎦⎣ ⎦⎩⎭Which of the above subsets of are spanning sets for ?3\3\6. (12 pts) Consider the following subsets of :3\i. ii.112, 224⎧⎫−⎡⎤⎡⎤⎪⎪⎢⎥⎢⎥−⎨⎬⎢⎥⎢⎥⎪⎪⎢⎥⎢⎥−⎣⎦⎣⎦⎩⎭1221, 1 , 1211⎧⎫−−⎡⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥−⎣⎦⎣ ⎦⎣ ⎦⎩⎭iii. iv. 1211, 1 ,0211⎧⎫−⎡⎤⎡⎤⎡⎤⎪⎪⎢⎥⎢⎥⎢⎥−⎨⎬⎢⎥⎢⎥⎢⎥⎪⎪⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⎩⎭12 11, 1 , 2122⎧⎫−⎡⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎣⎦⎣ ⎦⎣ ⎦⎩⎭Which of the above subsets of are linearly independent?3\7. (5 pts) Consider the set which is a subset of . Is the set S linear22{1 , 1 , 2 }Sxxxxx=− ++ −−3Pindependent?8. (5 pts) Consider the set which is a subset of . Is the set S linear2{cos2 ,cos ,1}Sxxππ=2(,)C −∞ ∞independent?9. (9 pts) Consider the following subsets of :2\i. ii. 11,33⎧⎫−⎡⎤⎡ ⎤⎨⎬⎢⎥⎢ ⎥−⎣⎦⎣ ⎦⎩⎭13,26⎧⎫−−⎡⎤⎡ ⎤⎨⎬⎢⎥⎢ ⎥−−⎣⎦⎣ ⎦⎩⎭iii. 23,46⎧⎫−−⎡⎤⎡⎤⎨⎬⎢⎥⎢⎥−⎣⎦⎣⎦⎩⎭Which of the above subsets of form a basis for ?2\2\10. (12 pts) Consider the following subsets of :3\i. ii.1231,0, 111 0⎧⎫⎡⎤⎡⎤⎡⎤⎪⎪⎢⎥⎢⎥⎢⎥−−⎨⎬⎢⎥⎢⎥⎢⎥⎪⎪⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⎩⎭1211, 1 , 2121⎧⎫−−⎡⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥−⎣⎦⎣ ⎦⎣ ⎦⎩⎭iii. iv. 11 21, 3, 2121⎧⎫−⎡⎤⎡⎤⎡⎤⎪⎪⎢⎥⎢⎥⎢⎥−−−⎨⎬⎢⎥⎢⎥⎢⎥⎪⎪⎢⎥⎢⎥⎢⎥−−⎣⎦⎣⎦⎣⎦⎩⎭0211, 1 , 2122⎧⎫−−⎡⎤⎡ ⎤⎡ ⎤⎪⎪⎢⎥⎢ ⎥⎢ ⎥−−⎨⎬⎢⎥⎢ ⎥⎢ ⎥⎪⎪⎢⎥⎢ ⎥⎢ ⎥−⎣⎦⎣ ⎦⎣ ⎦⎩⎭Which of the above subsets of forms a basis for ?3\3\11. (4 pts) Consider the following statement:Theorem. Let and let . The following are equivalent:{}123,,,,nn⊂"\aaa a[]123 nA = "aaa ai. A is invertibleii. has a unique solutionA = 0xiii. A is row equivalent to Iiv. has a solution for every A =xbn∈ \bv. det( ) 0A ≠vi. tr( ) 0A ≠vii. is a basis for {}123,,,,n"aaa an\viii. 123dim( ({ , , , , }))nspan n="aaa aix. () {}NA= 0 () {}NA= 0x. 0A ≠Which two of the above statements need to be deleted to make the theorem true?Name _________________________ Form BAnswers1. i. Yes No ii. Yes No iii. Yes No iv. Yes No2. i. Is Not Satisfied Is Satisfied ii. Is Not Satisfied Is Satisfiediii. Is Not Satisfied Is Satisfied iv. Is Not Satisfied Is Satisfied3. i. Yes No ii. Yes No iii. Yes No4.i. Yes No ii. Yes No iii. Yes No5. i. Yes No ii. Yes No iii. Yes No iv. Yes No6. i. Yes No ii. Yes No iii. Yes No iv. Yes No7. i. Yes No8. i. Yes No9. i. Yes No ii. Yes No iii. Yes No10. i. Yes No ii. Yes No iii. Yes No iv. Yes No11. ______,