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MATH 110 - Fall 2002 - Ribet - Midterm 2 (solution)

MATH 110 - Fall 2002 - Ribet - Midterm 2 (solution) - Math...

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Math 110 Professor K. A. Ribet Midterm Exam October 31, 2002 Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Please write your name on each sheet of paper that you turn in. Don’t trust staples to keep your papers together. Explain your answers as is customary and appropriate. Your paper is your ambassador when it is graded. In this midterm, the scalar field F will be the field of real numbers unless otherwise specified. These solutions were written by Ken Ribet. 1. Let A be the 100 × 100 matrix of real numbers whose entry in the ( i, j ) th place is ij for 1 i, j 100 . Determine the rank of A . (Figure out what it is and show that the rank is what you say it is.) Looking over the papers that were handed in, I got the impression that this problem was easy for most students. As people pointed out, the j th column is j times the first column, so the span of the set of columns is the span of the first column. This span is 1-dimensional, so the rank is 1. More generally, we can take two vectors ( a 1 , . . . , a n ) and ( b 1 , . . . , b m ) and form the matrix whose ( i, j )th entry is a i b j . This is called the
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