Math 636 Midterm Quiz \u001a\u0014 \u0015 \u001b x1 2 1. The geometric interpretation of the following set W = R | x1 + x2 = 2 is x2 (a) a point in R2 . (b) a line in R2 passing through the origin. (c) a line in R2 not passing through the origin. (d) all of R2 . 2. The of the set S of all vectors ~x R4 that are orthogonal to geometric interpretation 1 0 1 0 2 1 , , and is 1 1 1 1 1 2 4 (a) a point in R . (b) a line in R4 . (c) a plane in R4 . (d) a hyperplane in R4 . 3. Let vectors in R4 such that the solution set of the system \u0002 ~v1 , ~v2 , ~v3 , ~v4 be non-zero \u0003 ~v1 ~v2 ~v3 ~v4 ~0 is a hyperplane. What is the geometric interpretation of Span{~v1 , ~v2 , ~v3 , ~v4 }? (a) A point in R4 . (b) A line in R4 . (c) A plane in R4 . (d) A hyperplane in R4 . 4. Which of the following is a scalar equation for the plane in R3 with vector equation 1 1 1 ~x = s 1 + t 2 + 1 , s, t R 0 3 1 (a) (b) (c) (d) x1 x2 + x3 = 1 x1 + x2 + x3 = 1 3x1 + 3x2 + 3x3 = 1 x1 x 2 = 6 1 2 5. Which of the following is a vector equation for the plane in R3 with scalar equation x1 x2 +2x 3 = 0. 1 0 (a) ~x = s 0 + t 1, s, t R. 2 2 1 2 2 (b) ~x = 1 + s 0 + t 2, s, t R 1 1 0 1 2 (c) ~x = s 1 + t 2, s, t R. 2 4 1 1 2 (d) ~x = 1 + s 1 + t 0 , s, t R 1 0 1 6. Let (a) (b) (c) (d) 1 8 1 1 A= 1 1 3 3 det A = 24. det A = 84. det A = 24. det A = 84. 7 7 1 1 . Then 3 1 4 1 x1 3 x2 R | x1 + 2x2 = 3x3 , then 7. If T = x3 (a) dim T = 1. (b) dim T = 2. (c) dim T = 3. (d) T is not a subspace of R3 . 8. If U = Span{3 + x + 3x2 , 1 + 2x + x2 , 1 3x + x2 , x} is a subspace of P2 (R), then (a) dim U = 1. (b) dim U = 2. (c) dim U = 3. (d) dim U = 4. 3 9. There was a typo in the original question. The answer is (d). \u0014 10. Let A = (a) A B (b) A B (c) A B (d) A B \u0015 \u0014 \u0015 1 3 4 3 and B = . Then 2 4 1 2 is not \u0014 defined. \u0015 3 0 = 3 2 \u0014 \u0015 5 0 = 3 6 \u0014 \u0015 3 6 = . 1 2 \u0014 \u0015 \u0014 \u0015 5 2 11. The area of the parallelogram determined by ~x = and ~y = is 2 3 (a) 19. (b) 19 (c) 11 (d) 11. 12. If A is a 3 4 matrix and AB is a 3 5 matrix, then (a) B is a 4 5 matrix. (b) B is a 3 5 matrix. (c) B is a 5 3 matrix. (d) B is a 5 3 matrix. 4 2 1 1 1 1 , 2 , 0 . If [~x]B = 2 , then 13. Consider the basis B = 2 2 3 2 1 1 7 3 (a) ~x = 1 (b) ~x = 2 (c) ~x = 3 (d) ~x = 3 4 2 0 12 14. There was a typo in the original question. The answer is (d). 15. Which of the following is not an elementary matrix? 1 0 0 1 0 3 1 0 0 (a) 0 1 0 (b) 0 1 0 (c) 0 2 1 0 0 1 0 0 1 0 0 1 0 0 1 (d) 0 1 0 1 0 0 16. Let E1 , . . . , Ek be n n elementary matrices such that Ek E1 A = I. Which of the following statements is true. (a) There may exists a vector ~b Rn , such that the system of linear equations A~x = ~b is inconsistent. (b) For all ~b Rn , the system of linear equations A~x = ~b is consistent with unique solution ~x = Ek E1~b. (c) For all ~b Rn , the system of linear equations A~x = ~b is consistent with unique solution ~x = Ek1 E11~b. (d) For all ~b Rn , the system of linear equations A~x = ~b is consistent with unique solution ~x = E11 Ek1~b. 5 For questions 17 - 30, determine if the statement is True or False. You should make sure that you have a proof of each true statement and a counter example for each false statement. 2 1 1 1 3 2 , 5 , 2 . 17. In R , the vector ~x = 1 is in the span of 3 7 1 1 (a) True. (b) False. 18. If a system of 3 linear equations in 4 unknowns is such that the coefficient matrix has a rank of 3, then the system has infinitely many solutions. (a) True. (b) False. 19. Let ~u, ~v Rn . If k~u + ~v k2 = k~uk2 + k~v k2 , then ~u and ~v are orthogonal. (a) True. (b) False. 20. If A and B are n n matrices such that AB = 0, then either A = 0 or B = 0. (a) True. (b) False. 21. If A and B are n n matrices, then (A + B)(A B) = A2 B 2 . (a) True. (b) False. 22. If B 2~x = ~0 for some vector ~x 6= 0, then B is not invertible. (a) True. (b) False. 23. If A is an n n matrices such that A2 2A + I = 0, then A is invertible. (a) True. (b) False. 24. If a subspace S of Rn does not contain any of the standard basis vectors, then S = {~0}. (a) True. (b) False. 6 25. If {~u, ~v , w} ~ is a linearly independent set in a vector space V, then there exists c1 , c2 R such that ~u = c1~v + c2 w. ~ (a) True. (b) False. ~ 1~v1 + +ck~vk , c1 , . . . , ck R, 26. If the solution set of A~x = ~b has vector equation ~x = d+c ~ ~v1 , . . . , ~vk }. then the solution space of A~x = ~0 is spanned by {d, (a) True. (b) False. 27. For all non-zero vectors ~u, ~v R3 , the set {~u, ~v , ~u ~v } is linearly independent. (a) True. (b) False. 28. If R is the reduced row echelon form of an invertible matrix A, then det A = det R. (a) True. (b) False. 29. If A and B are row equivalent, then there exists an invertible matrix C such that A = CB. (a) True. (b) False. 30. If A and B are m n matrices such that rank(A) = m = rank(B), then A and B are row equivalent. (a) True. (b) False. \f