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Unformatted text preview: 8832832885 15:23 5188422415 LICE MnﬂiIN LIERﬂiR‘r" F'nﬂiGE 8l."'8E PROFESSOR KENNETH A. RIBET Second Midterm Examination November 2, 2005
2:103:00 PM Name: GSI’s Name: SID: Special Status (e.g., Math 49 or Concurrent Enrollment): Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players,
PDAs, and other electronic devices. You may refer to a single 2sided sheet of notes. Your
paper is your ambassador when it is graded. lClorrect an3wers without appropriate supporting
work will be regarded skeptically. Incorrect answers without appropriate supporting work
will receive no partial credit. This exam has six pages. Please write your name on each page.
At the conclusion of the exam, please hand in your paper to your GSI. 8832832885 15:23 5188422415 LICE MnﬂaIN LIERﬂR‘r" F'nﬂaGE 82."'8E 8 —1 1 U
6 —1 2 —3
C} —1 m3 —10
O O D O 1. Let R = . Exhibit bases for the following three spaces: DOOM u the row space of R,
o the column space of R1
1 the null Space of R. 8832832885 15:23 5188422415 LICE MnﬂiIN LIERﬂR‘r" F'nﬂiGE 83.385 3 0 w1
2. Find three linearly independent eigenvectere for the matrix 0 2 0 , whose
—1 0 3 characteristic polynomial is (A — 4)()\ _. 2)2 15 this matrix diagonalizeble? 8832832885 15:23 5188422415 LICE MnﬂaIN LIERﬂR‘r" F'nﬂaGE 84.385 3. Let W be the span of the three vectors v1 = (1,—1,3,—2), v3 = (1,9,1, —10) and
U3 5 2m  “Hg in R4. What is the dimension of W? Find an. orthogonal basis for W. UHIZHIZUUIJ 1'3: 22:! 2311:113422411: LILJH Mn'ﬂ'ulN LlﬁHIﬂuH‘r’ I‘n'ﬂ'ullsilz lathFUR: 1 O O 2 4 6 8
D 1 0 5 12 13 9
O (J 1 ml 31 5 23
4. Evaluate the determinant of the matrix 0 0 0 4 2 7 1
D 0 0 —2 1 3 —2
U U 0 0 1 0 0
U D O —1 2 5 3 UHIZHIZUUIJ 1'3: 22:! bllﬁb4224lb LILJH Mn'ﬂ'ulN LlHHIﬂuH‘r’ I‘n'ﬂ'ullsilz Ubﬂﬁb 5. Let A be an n x 71. (square) matrix. Suppose that A2 = A. Show that Ay = y for all y in
the column space of A. If the null Spams of A is {0}, show that A is the identity matrix of
size n. ...
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 Spring '10
 Budinger

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