This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: THE FACULTY OF ARTS AND SCIENCE University of Toronto W .
FINAL EXAMINATIONS, August 2003 Y MAT 223H1Y— Linear Algebra I Examiners: Agustin HerreraMorales Ching—Nam Hung Last Name: , Student #
———————————___ First Name: >
(a) TIME ALLOWED: 3 hours
(b) NO AIDS ALLOWED
(c) TOTAL MARKS: 100
(e) WRITE SOLUTIONS ON THE SPACE PROVIDED (f) GIVE ANSWERS FOR THE MULTIPLE CHOICE QUESTIONS ON
THE SECOND PAGE, ONLY THOSE ANSWERS WILL COUNT (9) DO NOT REMOVE ANY PAGES. THERE ARE 15 PAGES
(h) MAKING SCHEME: (1) Multiple Choice Questions : see page 3 (2) Written Questions: indicated by each question MARKER'S REPORT NAME: Student No Give answers for multiple choice questions below.
ONLY THOSE ANSWERS WILL COUNT Right way of giving the answer : . W
®®®@@© 1 

4—
7— ®®®@®© ®®®@©©
®®®@®©
®®®@®© 10'
11 12 PART I: 12 MULTIPLE CHOICE QUESTIONS
To each multiple choice question, there is only ONE correct answer. Marking Scheme: 4 marks for each correct answer. 0 mark for a wrong answer. ONLY the answers on page 2 will count.
1. Which of the following statements is FALSE. (1) If U 71$ 0 is a. subspace of R", it has an orthogonal basis. (2) X and Y in R" are orthogonal if and only if “X + YH2 = “XII2 + HY2.
(3) If ”X“ = 3, ”Y“ = 1, and [IX — Y“ = 3, then “X + Y“ = 3. (4) X  Y = 41{HX +YH2 — “X  Y2}~ (5) “X”2 + ”Y“2 = %{HX + Y“2 + ”X — YHZ} (6) Let U be a subspace of R". If P + Q = 0 with P in U and Q in Ui, then
P=Q=u 2. Which of the following statements is TRUE? (1) If A 2 CR where C 76 0 is a column in R“ and R 75 0 is a row in R", then
colA = span{C} and rowA = span{R} and nullA Q null R. (2) If nullA : 0 where A is a matrix, then A = 0. (3) If A is an n x 11 matrix, then A2 = 0 if and only if A = 0.
(4) If A is an n x 77. matrix, and A2 = 0, then rankA = 0. (5) colA is never equal to null A for any matrix A. (6) Let A be m x n and B n x m. Then colB = nullA if and only if AB = 0. so“). —a b 0
O O 0 a
0 0 —a p _
and are, respectively,
0 b q k a s t u
(1) ——a2b and (1%.
(2) ~a3b2 and (1%.
(3) azb and a3b.
(4) a3b2 and —a3b.
(5) azb and —a + b. (6) —b and —a + b. 4. Let T : R3 ——> R3 be a linear transformation satisfying condition T((l 1 0)T) = (1 0 0)T,T((1 0 —1)T)=(0 1 0)T,T((—1 1 2)T)=(1 —2 W. Then (1) T((a b c)T) can be determined by using the given conditions, for arbitrary a, b, c.
(2) It is necessary that T((l 1 1)T) = (2 3 — 1)T.
(3) T((—3 2 5)T) = (2 — 5 0)T.
(4) dim kerT + dim im T = 2.
(5) If, in addition, T((O 0 1)T) = (1 0 0)T, then T is onto. (6) T((a b —a+b)T) = (a b 1)T. £6“ 5, EU = span{(1 0 1)T, (0 1 1)T, (2 1 3)T}, and if (a b c)T is in UJ', then one of the following is the possibility for a and b. (3) a=b=tfor some t. (4) a=t2+1,b=t—2forsomet.
(5) a=2,b=t2+3forsomet.
(6) a=b2+1. 6. Let Pu be the set of all the polynomials of degree at most n, and F[a, b] the set of all the continuous functions on [a, b]. Then
(1) dimPn = n +1 and {1, sinx, cosz} is a basis of F[a,b].
(2) {1, (x — a), (x — a)2, (m — a)3, . . . , (a: — a)"} is a basis of P,1 for every a.
(3) If f(x) is in F[a,b], then f(a) = f(b).
(4) {f(z) E F[a, b] : f(a) = f(b) 95 f(lgiﬂ is a vector subspace of F[a, b]. (5) {p E Pn : p(—1) = 0 and p(1) = 0} need not be n — 1 dimensional vector subspace of P”. (6) {:122 + 1, m2 + x} spans a 3 dimensional subspace of Pn because 1, x, :32 appear in the two polynomials. 90$6 .. 7. Let A, B, C be square matrices. Which of the following statements is FALSE?
(1) If AB = 0, then (BA)3 2 0.
(2) If AB = 0 and if A 95 0, then I — B is invertible.
(3) If AB is invertible, then A and B are both invertible.
(4) If AB and BA are both invertible, then A and B are both invertible.
(5) A, C and ABC are all invertible, then B is invertible.
(6) AB = A0 implies B = 0 when A is invertible. 8. Let '0 = (1, 1 0):", U = span{(1 ,0, 1)T, (1 ,1, —1)T}. Then ProjU(v) is equal to
(1) (1 2, 1)T. (2) (—3/2, 0, —2/3)T.
(3) (7/6, 2/3, —1/6)T.
(4) (—1/6, 1, 0)T.
(5) (1, ~1/3, —1)T. (6) (0, ~1, —1/2)T. 9. If {X , Y, Z} is a basis of R3. Which of the following is not a basis of R3?
(1) {X—Y,X+Y, Z}.
(2) {2X—Z, Y—Z,X+Y+Z}.
(3) {X+Y+Z,X—Z,Y—Z}.
(4) {X+Y—Z,X+Y+Z,Z}.
(5) {X,Y+Z,X+Z}. (6) {X+2Z,X+2Y,X—2Y+Z}. 1 1 1
10. A = < >, then (nullA)i has an orthogonal basis
2 3 1 (1) {(1 ,2, 0)T,(1, 0, 2)T} 1 1
) . Which of the following statements is TRUE? 11. LetA=(
a 1 (1) If a = 0, then A814 is diagonalizable.
(2) If a = 2003, then A is diagonalizable.
(3) A is diagonalizable for every value of a.
(4) If a : 1, then A is not diagonalizable.
(5) If IA} aé 0, then A is diagonalizable. (6) If a = 1, then A25 is not diagonalizable. 1 2 —1 1 ——2
1 2 1
12.LetA= 21 1 ,B= 1 2 ,0: ,andletTA,
—1—1 —1
1 5 —4 2 1 TB, To be the linear transformations induced by A, B, C respectively. Then
(1) TA, T3, T0 are 11.
(2) TA, TB, TC are onto.
(3) ker To = {O}, the trivial subspace of R3.
(4) im T3 = R3.
(5) ker TA aé {0}, im TB is 3 dimensional subspace of R3, im T0 = R2. (6) (kerTA)i=span{(1, 0, 1)T,(1, 1, 0)T} PART II: OPEN ENDED QUESTIONS Marks for each question are indicated to the right of the question number. Full mark will be given only if all the steps in the solution are justiﬁed and the solution is clearly presented.
3 1 l
2 0 1 . .
13(a).[8 marks] Let A = . Find a hams of null A and a. basis of im A,
4 2 1
1 ——1 l and determine rank A. 13(b).[4 marks] Let S : R" —+ R” and T : R” —> R" be linear transformations . Show that if S and T are both 11 (or onto), then m = n. 13(0).[5 marks] Let A be m x n and B n x m where m > n. Determine whether matrix AB is invertible or not, and justify. 10 1 4 2
14(a).[8 marks] Let/1: 0 —3 4 0 4 3
matrix D such that P‘IAP = D. . Find an invertible matrix P and a diagonal 11 14(b).[5 marks] Let A be a square matrix. Prove: A has an eigenvalue A = 0 if and only if IA] 2 0, where [A] is det A. 14(0).[5 marks] Let {X1,X2,X3} be an orthogonal set of nonzero vectors in IR".
Prove that {X 1, X2, X3} is linearly independent. 00666". 12 15(a).[6 marks] Suppose A is an m x n matrix and B is a column in R“, and consider a
column vector Z in R". Show that if Z satisﬁes (ATA)Z = ATB, then AZ = ProjU(B)
where U = {AX : Xin R"}. (Hint: You may assume that if v = P + Q where P e U
and Q 6 UL then P = ProjU(v).) 0““‘6... 13 15(b) [6 marks] Let P3 be the set of all the polynomials of degree at most 3. Let
U = {p 6 P3 : p(1) = 0} be a vector subspace of P3. Find a basis of U. code 14 d... 15(0) [5 marks] Let V be a vector space. Suppose u, v, w are 3 nonzero vectors in V
satisfying {u, v} is linearly independent, {u,w} linearly independent, {11,20} linearly
independent. Does it necessarily imply that {u, v, w} is linearly independent? If yes, prove it. If no, provide a counter—example. 15 ...
View
Full Document
 Spring '09
 UPPAL
 Linear Algebra, Algebra

Click to edit the document details