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**Unformatted text preview: **EXAM 2
Math 51, Spring 2004. You have 2 hours. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING
TO RECEIVE CREDIT Good luck! Name go! 1.) it‘ms ID number 1. (/30 points) 2. (/30 points) 3. (/30 points) 4. (/30 points) 5. (/30 points) Bonus (/15 points) Total (/ 150 points) “On my honor, I have neither given nor
received any aid on this examination. I
have furthermore abided by all other aspects of the honor code with respect to
this examination.” Signature: Circle your TA’s name:
Brett Parker (2 and 6)
Chad Groft (3 and 7) Joe Blitzstein (4 and 8) Ryan Vinroot (ACE) Circle your section meeting time: 11:00am 1:15pm 7pm 1. Find the determinant and the inverse of the matrix 10000 01020 00103 7—2010 00001
I O o 0 O l O O O O
O | o 2 O o l O O O
0 (j | O 3 O O I o o
7 —Z O | O o o O I o
O O 0 0 1 o C) o (3 I
l O O o 0 I O o o 0 V:
O l o 2 o o \ O o C) [‘2
O O \ O 3 o O l O 0 r3
0 O O 5— O ‘7 2 O ( 0 111 +28
0 O O O l o o O O f r S— f O o o O I C) O Q 0 r1
0 ( a 0 0 Isl/g. 1/; O _%_O r1—%n+
O O I O C) O O ‘ O ’3 {aviary
O O O ' 0‘7; 7/5 o 1/; o C4/5"
O O O o( C) O o O | F5— 3. Find the equation of the ellipse obtained by rotating (clockwise around the origin by angle
7r/ 6) the ellipse with equation I Hint: A point [y on this new ellipse is deﬁned by the property that, if rotated counter- u clockwise around the origin, the resulting point [v :| will be on the original ellipse and thus will satisfy the equation Lzlr T CDU‘n‘ief‘ OLQQLJISQ arbqu ~6- n76:
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*‘3 may (if) “’7 A v2, 5/2 4. Let A be an m x 71 matrix. Describe the precise procedure by which you would determine bases for the column Space and the null space of A. Based on this description, preve the
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§;nce J’L-E WUMLLF‘ 0‘; QL/nxm‘l‘: HA 0k (RS—[S IS (Mmglm/ 5. Prove that for every linear transformation T : R” —> Rm, there exists a. matrix A such
that for 3.11 E’ e R”, Tm = A? QiVﬁ‘A ‘Hva hmar iW§LrMLM T, DQQ'Qi‘m‘i 0“ Mlﬂk A L7 A: TL) chm “3 3‘“ O‘H‘W W5 5, 'l'Lﬁ CJumnS (A: A 191 £9.9th as ‘LLG T0?) = A? 6 Bonus Question: Find a matrix A such that ( 1) None of $1,142,143, . . . ,A9 is the zero matrix
(2) A1” is the zero matrix. F Lei T a a m mama Md L7 A7 :TOWT .. lo a; 4 H) a
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(ABQLWHV, M m m A 413 Lager W} Me the me .,5
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