MA554_AUG10

# MA554_AUG10 - Math 55400 Qualifying Exam August 2010 W...

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Unformatted text preview: Math 55400 Qualifying Exam August, 2010 W. Heinzer PUID: Instructions: 1. The point value of each exercise occurs to the left of the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 14 5 16 6 14 7 16 8 20 9 14 10 12 11 14 12 12 13 16 14 12 Total 200 1 Math 55400 Qualifying Exam August, 2010 W. Heinzer 1. (20 pts) Let T : V V be a linear operator on a finite-dimensional vector space V and let R = T ( V ) denote the range of T . (a) Prove that R has a complementary T-invariant subspace if and only if R is independent of the null space N of T , i.e., R N = 0. (b) If R and N are independent, prove that N is the unique T-invariant subspace of V that is complementary to R . 2 Math 55400 Qualifying Exam August, 2010 W. Heinzer 2. (20 pts) Let V be a 5-dimensional vector space over a field F and let T : V V be a linear operator. (a) Prove that V is the direct sum of its two subspaces Ker T 5 = the null space of T 5 and Im T 5 = T 5 ( V ), the range of T 5 . (b) Give an example of a linear operator T such that V is not the direct sum of its subspaces Ker T and Im T . 3 Math 55400 Qualifying Exam August, 2010 W. Heinzer 3. (14 pts) Let n be a positive integer, let V be an n-dimensional vector space over a field and let T : V V be a linear operator. Prove or disprove that rank T + rank T 3 2rank T 2 ....
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MA554_AUG10 - Math 55400 Qualifying Exam August 2010 W...

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