midterm2

midterm2 - Harvard University Midterm 2 for Math 121 Fall...

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Harvard University Midterm 2 for Math 121, Fall 2007 Wednesday, November 21, 2007 Time allowed: 53 minutes 3 pages (including this one) Last Name: First Name: Harvard ID Number: Apart from question 1, you must fully justify your answers. On this exam the full score is 100 points. You may assume all vector spaces are ﬁnite-dimensional unless otherwise stated. Recall that for a linear transformation T : V W , the notation N ( T ) denotes the null space and R ( T ) denotes the range. Good Luck! 1

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Harvard University Midterm 2 for Math 121, Fall 2007 Wednesday, November 21, 2007 1. (20 points) Mark the following statements true or false. No justiﬁcation is needed. a) The rank of a triangular matrix is equal to the number of the diagonal entries diﬀerent from zero. b) There exist A,B M n ( C ) with AB nilpotent and BA not nilpotent. ( N M n ( C ) is called nilpotent , if there exists a positive integer m with N m = 0.) c) For all A,B M n ( C ) it is: rank ( AB ) = rank ( BA
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midterm2 - Harvard University Midterm 2 for Math 121 Fall...

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