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: Faculty of Mathematics QXV ﬂ University of Waterloo
Maw/t V“ MATH 136 — Winter 2003 FINAL EXAMINATION Date: April 167 2003 NO AIDS, NO CALCULATORS
Time: 9:00 — 12:00
Instructors: (Please check your section)
Section 1 K. Zhao
2 C.Y. Lam
3 K. Zhao
4 F. Szechtman
5 K.J. Giuliani 6 CG. Hewitt DDDDDDD 7 H. Muzaffar Family Name: (please print) _____________________.___ Initials: Id. Number: Signature: _____________________ This exam has 14 pages, please check to make sure
that you have them all. Your mark will be affected by how clearly you
present your solutions. ‘ Math 136 — Final Exam Winter 2003 1. a) Find conditions on the parameter p E R for Which the system of equations 1.m+2.y+3.z = ——1
2.x+5.y+(p2~—13)z=p+7
2.$+4.y+(p2— 11)z=p+3 has (i) No solutions (iii) A unique solution (iii) More than one solution. (i) No solution for: (ii) A unique solution for: (iii) More than one solution for: b) Find the rank and nullity of A and AT when “>me
mAww
CbU‘rhC/O
«lama;
0040501 Page 2 of 14 Math 136  Final Exam Winter 2003 2. Let S = {V1V2, . . .vp} be a subset of a vector space V. a) Deﬁne what it means for S to be linearly independent. b) Deﬁne Span(S). c) Deﬁne what it means for S to be a basis for V. (1) Let v 6 V and suppose that S is a basis for V. Prove that v may be written as
a unique linear combination of the elements of S. Page 3 of 14 Math 136  Final Exam . Winter 2003 1
3. Let A be the matrix A = 0
1 a) Evaluate A‘l. b) Express A as the product of elementary matrices. Page 4 of 14 Math 136 — Final Exam Winter 2003 4. In P2(R) consider the two bases 3 = {t2 + 1,1? — 2,t + 3} and T = '{2t2 + t,t2 + 3, t}. a) Find Ps._T, the transition matrix from T to S. You may use Q3a if you Wish. b) If v = 8t2 —— 4t ﬁnd MT. 0) Use your solution to a) to evaluate [v]5. Page 5 of 14 Math 136 — Final Exam Winter 2003 5. Let A,B e ManOF). a) Prove that if A and B are invertible then AB is invertible. Let C E MnxnﬂF). We say that C is nilpotent if 0’“ = On for some positive integer
k. Note that 0’“ = 0.0. . . C (k times). 0 1 1
b) Show that C: 0
0 0 1 is nilpotent.
0 0 c) Prove that every nilpotent matrix is noninvertible. Page 6 of 14 Math 136 — Final Exam Winter 2003 6. Let V = nggﬂR) and B E V is a ﬁxed matrix. Let 53 = {A E V : AB = 2BA}. a) Prove that $3 is a vector subspace of V. b) Give a B such that 33 = V. c) Give a B such that $3 = {(3 3) }. Page 7 of 14 Math 136  Final Exam Winter 2003 7. DBIBIIIHHB Whether or not the given function is an inner product on the given vector space. a) V = R2,<(x1>, <91>>= x? + mg + y? + y§ where x1,x2,y1,y2, E 1R 132 112 z w _ _
b) V = C2, <( 1) , (“13> = zlwl + 222112 — zlwz — 221111 where 21, 22,w1,w2 E C. (I;
_ c) V = 1R2, <<m:) , (39> = 2:131:11 +4z2y2+2m1y2+2x2y1 where 11:1, $2,311, 112 6 R Page 8 of 14 Math 136  Final Exam Winter 2003 8. a) Let (V, <, >) be an inner product space and W be a vector subspace of V. Deﬁne
WJ'. b) Find an orthonormal basis for the subspace, S, of R3 given by
S = {(23, y, z) E R3 : a: — 2g + 3z = 0}. Use the standard inner product. c) Find an orthonormal basis for 5*. Page 9 of 14 Math 136 — Final Exam Winter 2003 9. Consider the vector space of continuous real valued functions deﬁned on the interval
[—1, 1], V = C'[—1, 1]. Let the inner product on V be given by <f<t>,g<t>> = /—1f(t)g(t)dt' Given that B = { «i5, cos 7rt, sin m} is an orthonormal basis for a subspace S of V, use
the following information to ﬁnd: (i) the projection of the function f (t) = 132 onto S, and
(ii) ﬁnd the length of this projection. 1 1 1
Given Information: / tzdt = 2/3, / t2 cos 7rt dt = —4/7r2, / t2 sin 7rt dt = 0. —1 —1 —1 Page 10 of 14 Math 136  Final Exam Winter 2003 10. Consider the following 5 data points:
(—2,0), (~1,1), (0,1), (1,0), (2,1). Use the method of least squares to determine the equation of the line which is the best
ﬁt to the given data. That is, ﬁnd a, b, E R such that y = a3: + b is the best ﬁt to this
data. Page 11 of 14 Math 136 — Final Exain Winter 2003 11. let S = {V1,V2, . . . VP} be an orthogonal set of nonzero vectors in an inner product
space (V, <, >). Prove that S is linearly independent. Page 12 of 14 Math 136  Final Exam Winter 2003 12. Let V be a ﬁnitedimensional vector space and (V, <, >) be a real inner product space.
Suppose {V1, v2, . . . ,vp} is an orthonormal set of vectors in V and x E V. :2
Prove that H x ”22 Z(x,vi)2.
i=1 Page 13 of 14 ' Math 136 — Final Exam Winter 2003 ROUGH WORK Page 14 of 14 ...
View
Full Document
 Winter '08
 All
 Linear Algebra, Algebra

Click to edit the document details