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Unformatted text preview: Surname: ”—1 First Name: %_—
se Print Faculty of Mathematics MATH 136 Monday 14 March 2005 Question Mark Winter 2005 Id. No.: ﬂ. University of Waterloo
MIDTERM EXAM #2 19:00 — 20:15 Instructions: 1. 2. Only Faculty approved calculators are
permitted. The exam has 5 problems. Unless oth—
erwise indicated, you must Show all of
the work that you did to obtain a solu—
tion. . You may use any result from class with— out proof, unless you are being asked to
prove this result. Make sure to clearly
state any result from class that you are
using. . Make sure to put your name and ID number at the top of this page. . If you do not have enough space to ﬁn— ish your solution to a problem, continue
it on the back of the page, and indicate
that you have done so. . There are 2 blank pages at the end of the exam that can be torn off and
used for rough work. This exam has 10
pages in total, including the cover and
two blank pages. CIRCLE YOUR SECTION AND
INSTRUCTOR BELOW: Sec. Instructor Time
001 K. Giuliani 9:30
002 I. VanderBurgh 9:30
003 R. Levene 10:30
004 K. Giuliani 11:30
005 H. Wei 12:30
006 C. Hewitt 9200 l007 J. Muir 1:30 l A MATH 136, Midterm #2 1 2
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M m \000 MATH 136, Midterm #2 Page 4 of 10 Name: Hz \3 [3] 3. (a) Let V and W be vector spaces over 15‘, and T : V —> W a function.
Deﬁne what it means for T to be a linear transformation. l? T E Q \‘m ‘WOX‘FQ, W \l‘ da‘x‘iims Q soj Q? “$13
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" Q3) T La (‘0 == J01) a Teimrcmev ab‘xei closed MW 0603‘ E snowy woo“: (b) Deﬁne T : P1(R) —> R3 by T(a + bt) = (a + 2b, 2a — b, b).
(1P1 (IR) is the set of polynomials in t with real coefﬁcients and of degree less than or equal
to 1.) [4] i. Prove that T is a linear transformation. 0) new: um comm m7 a
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= (c (mm , e um)», are» ﬂ?“ MATH 136, Midterm #2 Page 5 of 10 Name'   \ (b) (continued)
Deﬁne T : EUR) —> R3 by T(a + bt) = (a + 2b, 2a —— b, b).
(1P1 (R) is the set of polynomials in t with real coefﬁcients and of degree less than or equal to 1.)
[4] ii. Determine a spanning set for the kernel of T. Is T 1—1? Explain. (Gd(Zia, ZG’lOi b» 7'" Cox Dub) gnu, WT: i R€P\L®\AR :3“): Q +2\O=O Q=Q
20’b=0 gt. 20=C3 [4] iii. Determine a spanning set for the range of T. Is T onto? Explain.
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C) + b MATH 136, Midterm #2 Page 6 of 10 Name' b [3] 4. (a) Let V be a vector space over F and H a non—empty subset of V.
Deﬁne what it means for H to be a subspace of V. ‘90? \\ \b be. cx eubspm, GbV M‘Qmm'wg
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(1% (R) is the set of polynomials in t with real coefﬁcients and of degree less than or equal to 2. p’ is the derivative of the polynomial p.)
[3] i H = {p E V l 13(1) + p’(1) =1}
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If there exists a b E W is such that T(x) = b has exactly one solution, prove that kerT = {0} (that is, the kernel lof T is the set containing only 0).
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