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09win-m1sols - Math 51 Winter 2009 Midterm Exam I Name...

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Math 51 - Winter 2009 - Midterm Exam I Name: Student ID: Select your section: Penka Georgieva Anssi Lahtinen Man Chun Li Simon Rubinstein-Salzedo 02 (11:00-11:50 AM) 03 (11:00-11:50 AM) 12 (1:15-2:05 PM) 17 (1:15-2:05 PM) 06 (1:15-2:05 PM) 11 (1:15-2:05 PM) 08 (11:00-11:50 AM) 21 (11:00-11:50 AM) Aaron Smith Nikola Penev Eric Malm Yu-jong Tzeng 09 (11:00-11:50 AM) 14 (1:15-2:05 PM) 15 (11:00-11:50 AM) 51A 20 (10:00-10:50 AM) 24 (2:15-3:05 PM) 23 (1:15-2:05 PM) Signature: Instructions: Print your name and student ID number, print your section number and TA’s name, write your signature to indicate that you accept the honor code. During the test, you may not use notes, books, calculators. Read each question carefully, and show all your work. There are nine problems on the pages numbered from 1 to 9, with the total of 100 points. Point values are given in parentheses. You have 2 hours (until 9PM) to answer all the questions. In the exam all vectors are columns, but sometimes we use transpose to write them horizon- tally. Thus v = v 1 v 2 . . . v k = [ v 1 , v 2 , . . . , v k ] T . Similarly v T is a row [ v 1 , v 2 , . . . , v k ]. The dot product of two vectors is denoted as v · w .
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Problem 1. (10 pts.) Mark as TRUE/FALSE the following statements. If a statement is false, give a simple example. If a statement is true, give a justification. a) The null space of the matrix 0 0 0 0 0 0 is R 2 TRUE FALSE The null space N ( A ) is a subset of R n , where n is the number of columns of A . b) The cross product of two vectors belongs to the plane spanned by them. TRUE FALSE The cross product is perpendicular to the vectors, not in the span. c) Let A be a 2 × 4 matrix. Then dim N ( A ) 2. TRUE FALSE dim N ( A ) is the number of columns without pivot. There is at most one pivot in each row, and there are only 2 rows, so at least 2 of the four columns will be without pivot. d) For any k × n matrix A , dim N ( A ) + dim C ( A ) = k . TRUE FALSE Sorry, it is n , the number of columns, not k . e) Span { v 1 , v 2 } is always a two dimensional linear subspace. TRUE FALSE It is true only if the two vectors are linearly independent. Otherwise the dimen- sion of Span { v 1 , v 2 } is smaller than two.
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