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Unformatted text preview: Math 107: Linear Algebra and Differential Equations Test #1 Name: I Thursday, October 1, 2009
' All answers must be justiﬁed. Question 1. [10 points] (a) Find the inverse of the following matrix (b) Use the inverse to compute the solution to the system of equations 43—23; = 5
2:12—3y = )(zvi
we Question 2. [16 points] Suppose that A and B are two 71 X n matrices with det(A) = 3 and def; (B) = 7.
Compute the following: (a) det(ABA) z 3x1}x57 63‘ (b) det(A—1) = 1/3 Question 3. [20 points] Consider the system of questions with unknowns 3'31, 352,” 3332 (13:1 + .713 = 1 — (1)332 + 2163 = 4
20ml + (b — a)32 + 323 = 2 (a) Determine the conditions on a and b so that the above system of equations admits a unique solution. a o [it a 0 I?! O 6 If!
. i 0 2:1}(2’5 O L—dk [‘2 A 0 5‘“. (:2. 2a 5..“ I}? CKDDlD a a 0‘0 (b) Determine the conditions on a and b so that the above system of equations has @ solution. baa, Question 4. [10 points] Under what condition on the constants a and b is the vector [Z] in the linear span of the vectors: a * (I
. "310‘ [2H
Zzuh 245t0kee'0'glkaa
' ’9 o—{OzloJr
746m YME I
' (Zizi
o—Bt 0L"2
'49 O o o: /?/Z(0\‘23 Question 5. [14 points] Given OH? 1%.? Wage 9 dfmﬁNSmD 52L
Two [QR/of? —? db :2. Question 6. [15 points]
(a) Show that the set satisfies the conditions for being a. subSpace of R3. (91+ (Mimi/IS ‘5) (x20, gab) C2) CCmecl W @0106?th Question 7. [20 points] (8.) Determine whether HE} [E]? [52]} form a spanning set for R3. Explain your answer. a l e—f IX 63f?
5: C g "7
Wot (“HQ/c3 “mad [c] i £31625 ’1' g g __H‘C O Q—ZZ‘C‘30K
ﬂ . a t a“ —+ I Back 0 o 45ngc—3a’Q—2a)?‘ «jer {mad qbw‘ciéé (Levee/“(fan
V 1am wgPammﬂ 56” 7%” W: (10) Does V above form a basis for R3? Why or why not. Yaglo (WWBVEICJFDFS ...
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This note was uploaded on 04/25/2010 for the course MATH 107 taught by Professor Trangenstein during the Fall '07 term at Duke.
 Fall '07
 Trangenstein
 Differential Equations, Linear Algebra, Algebra, Equations

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