Sample

Math 1Z2C3 Sample Exam Name: (Last Name) (First Name) Student Number: Tutorial Number: This exam consists of 44 multiple choice questions worth 1 mark each (no part marks). All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Letu, v, and w be independent vectors in R3. Which of the following sets are also independent? D{u—-v,v—w,w—u} () {u,u+v,u+v-+w} (i) {u —v,v—w,u+v+w, w} (a) () only (b) (ii) only (c) (i) and (ii) only (d) all of them (e) (ii) and (iii) only 2. The following set of vectors {(1,—1,1,-1),(2,0,1,0),(0,—-2,1,-2)} (a) Spans R* but is not independent. (b) Is independent, but does not span R* (¢) Is independent and spans R* (d) Is not independent, and does not span R* (e) None of the above. 3. Suppose that W = span{vy, v3, v3} where each v; is in R3. Consider the following statements. (1) If xis in W then x = ¢yv; + covy + c3vs for some scalars ¢y, co, cs. (1) 3vy — 2vy isin W. (iii) W is a subspace of R3. Which of the above statements are always true? (@) () only (b) (ii) only (c) (i1) and (ii1) only (d) (i) and (ii) only (e) (1), (i), and (iii)

4. Letu = (0,—2,2) and v = (1,3,—1). Which of the following vectors are in span{u, v}? (1) (1a _]-a 2) @i (1,1,1) (iii) (5, 3,7) (a) all of them (b) (i1) only (c) (iii) only (d) (i) and (iii) only (e) (ii) and (ii1) only 5. If {v, w} is independent, find conditions on the scalars k; and k; so that the set {k1v + w, v + kow} is also independent. @kithk=1 Mkit+thk#l ©k#k Dkk =1 (€kk #1 6. Let p = 2 — z + z?. Find the coordinates of p with respect to the following basis of P; {1+=2,1+ 22 z+ 2%}, (@) (17 —1, 2) (b) (07 2, _1) © (07 2, 2) (d) (27 —1, 0) () (_17 1, 3) 7. Let V' be a vector space with dimension . Consider the following statements. (1) Every independent set in V' is a basis for V' (i1) Every set in V' that spans V must be independent (i11) Every set in V' with less than n vectors must be independent. Which of the above statements is always true? (a) (i1) and (iii) only (b) (iii) only (¢) (i1) only (d) none of them (e) (i) only 8. Find the dimension of the following vector spaces. (1) The set of all 2 x 2 skew-symmetric matrices (ii) The set of all polynomials a + bx + cx® where a = b + c. (a)land2 (b)2and3 (c)land3 (d)3and3 (e)4 and2 9. If Aisa4 x 4 matrix and the columns of A are linearly dependent then, (a) every vector b in R* is in the column space of A (b) no vector b is in the column space of A (¢) The column vectors of A form a basis for R* (d) None of the above 10. Letu = (1,—2,1,6) in R% and let W = span{(1,1,—1,0),(1,1,0,0)}. Compute projy-u. (a)( %a(),]-a%) (b) (_ ’ ;7%70) (C) (__ _17170) @) (—3,—3,1,0) © (3,—1,—3,0)

17. 18. 19. 20. 21. For Questions 17-19, determine which of the following answers is correct for the given subset W of R3, (a) W is a subspace (b) W is closed under addition, but not closed under scalar multiplication (c) W is closed under scalar multiplication, but not closed under addition (d) W is not closed under scalar multiplication, and not closed under addition W = all vectors of the form (a, 3b, ¢) where a = ¢ + 1. @ (b) () (@) W = all vectors of the form (2a, —b?, —c) (@ () () d Let b be a nonzero vector in R* and let A be a 4 x 4 matrix. Let W = all vectors x in R* that are solutions to the equation Ax = b. (@ (b) (o) (d) Suppose that W is a subspace of a vector space V. Consider the following statements. (1) Ifuis in W and au — bv is in W (where b # 0) then v is in W. (i) Ifuisin W and visin W then au — bvisin W. Which of the above statements are always true? (a) (i) only (b) (1) only (¢) (1) and (ii) (d) neither of them Let W = span{(1,1,1,1), (3,1,3,1), (6,2,4,0)}. Find an orthonormal basis of W using the Gram-Schmidt process. @ (413 Dk b =D L) (b) {(27% % %) (07_%717_%)7 %707%7_%)} (©) {(27% % %) (Oa%a—%fl)),(fi,oao’—fi)} @ {3331, (B0 —52,0,0), 0,0, 15, 1)} (e) {(%7%’%7%)’(_%&’fiaox(o’%aovfi)}

22. Consider the following set of orthogonal vectors, vi=(1,-1,2,-1),vo =(-2,2,3,2), vs = (1,2,0,—1), v4 = (1,0,0,1). Letu = (3,1,—2,4). Find ¢ such that u = av; + bvs + cvs + dvy @2 s ©3 D35 (©3 23. If A and B are both n X n invertible matrices, which of the following matrices is the inverse of (A~1B)T? @ (B'A)" M (AB™H)" () B'(A")" @ (A")'B" (¢)(B'AT)! 24. Consider the following statements. (i) If u and v are orthogonal in R3 then |ju + v|| = |ju — v|| (i) |[u]|® + [|v]* = L[ja+ || + i|lu — v||* for all u, v in R®. Which of the above statements is always true? (a) neither (b) (i) only (c) (ii) only (d) (1) and (ii) 25. Recall that B is similar to A if there is an invertible matrix P such that B = P~1AP. Suppose that B is similar to A. Consider the following statements. (1) A and B have the same determinant (ii) B~ is similar to A~} Which of the above statements are always true? (a) (i) only (b) (i) only (c) (1) and (i1) (d) neither of them 26. A matrix P is called orthogonal if PPT = I. Consider the following statements. (1) If P is an orthogonal matrix then 2P is also orthogonal. (i1) If P is an orthogonal matrix thendet P = + 1 Which of the above statements is always true? (@) ) ony (b) (i) only (c) (i) and (i) (d) neither 0 0 a 27.Let A= [0 b O (. Find the characteristic polynomial p()) of A. a 0 O @ (A +b)(A +a)® (b) (A = b)(X + a)? © (A =bd)(A —a)’ A=A —a)A+a) @A+bd)A—a)A+a)

3S. 36. 37. 38. 39. Consider the following matrices. a=|1 o) =8 7 B can be obtained from A by the following sequence of row operations on A: 1. Switch row 1 and row 2 2. Replace row 2 by (row 2 — 2 X row 1) Using the above sequence of row operations (in the above order), find an invertible matrix U such that UA = B. (a)li :;] ®) [(1) —12] (© [(2) —14] (d)[_zl —34] (e)[_11 —32] Compute the determinant of the following matrix. (0 1 -1 0] 3 0 0 2 01 2 1 5 0 0 7 (@)0 (b)5 (c) —33 (d) —17 (e)8 Let A be a 2 x 2 matrix, with det A = 2. Evaluate det(2adj(A)). @2 ()4 (©)8 (d)16 (e) 32 A square matrix P is called idempotent if P? = P. Let A and B be n x n idempotent matrices. Which of the following matrices are always idempotent? i)A-—B (i) AB (a) ) only (b) (ii) only (¢) (i) and (i1)) (d) neither In a dynamical system for inheritance, suppose that the transition matrix has eigenvectors x; =(1,2,1), xo =(1,0,—1), and x3=(1,—2,1), with corresonding -eigenvalues M=1,)= %, and A3 = 0, respectively. If the initial state vector vy can be written as Vg = %xl — %xz + %X3, find the constant b so that the state vector after 5 generations can be written as v = axj + bxy + cXxj3. @ -1 ®-35 ©zx @5 gy

40. Let z be a complex number. Which of the following statements is correct? (@) Z + 2, (Z — 2)i, Zz are all real numbers (b) Z+ 2z, (Z — 2)i, zz all have modulus 1 (¢) Z + z and Zz are real numbers, but (Z — 2)i is not a real number. (d) If z is a complex number and |z| = 1,then z = 1 or z = —1. (e) none of the above 41. Find all complex numbers z so that 23 = —8s. @) \/3+4,—3+14,-2 ®)V2—i,-2—1i0,2 (©)/3—14—/3+i,—2 A V2+i,—/2+4,2i (€3—1i,—/3—1i,2 42. Find a matrix P which diagonalizes D= ] (b) [i _13] (c)[‘l1 }] @ | }] © [(1, f] DN (2) [‘1 43. Let A be an n X n matrix. Suppose that there exists an invertible matrix P such that P~ 'AP = D, where D is a diagonal matrix. Consider the following statements. (1) A2 — P2D2(P—1)2 (i) A2 = P~1D?P Which of the above statements is always true? (a) ) only (b) (i) only (¢) (1) and (i1)) (d) neither 44. The arrival of a bus at a particular stop can be classified as either an early arrival or a late arrival. If it is early on one day, then there is a 60% chance that it will be early the next day. If it is late on one day, then there is a 90% chance that it will be late the next day. In the long run, what proportion of times is the bus late? @2 M5 ©35 @ (@