MA554_AUG10

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 14

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online