MA554_JAN03

MA554_JAN03 - Math 554 (12) Qualifying Exam Heinzer January...

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

View Full Document Right Arrow Icon
Math 554 Qualifying Exam Heinzer January 7, 2003 (12) 1. Let F be a field, let n be a positive integer, and let W = F n × n denote the vector space of n × n matrices with entries in F . (i) Let W 0 denote the subspace of W spanned by the matrices C of the form C = AB - BA .Wha ti sd im W 0 ? (ii) For A F n × n , define the adjoint matrix adj A F n × n . (iii) If A R 3 × 3 and det A =2 ,whatisdetadj A ? (6) 2. Let T : C 5 C 5 be a linear operator and let g ( x ) be a polynomial in C [ x ]. If c is a characteristic value for g ( T ), must there exist a characteristic value a for T such that g ( a )= c ? Explain why or why not. 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 (20) 3. Let A C 4 × 4 be a diagonal matrix with main diagonal entries 1 , 2 , 3 , 4. Define T A : C 4 × 4 C 4 × 4 by T A ( B )= AB - BA . (i) What is the dimension of the null space of T A ? (ii) What is the dimension of the range of T A ? (iii) What are the characteristic values of T A ? (iv) What is the minimal polynomial of T A ? (v) Is T A diagonalizable? Explain.
Background image of page 2
3 (16) 4. Let F be a field, let m and n be positive integers and let A F m × n be an m × n matrix. (i) Define “row space of A ”. (ii) Define “column space of A ”. (iii) Prove that the dimension of the row space of A is equal to the dimension of the column space of A .
Background image of page 3

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

View Full DocumentRight Arrow Icon
4 (16) 5. Let D be a principal ideal domain and let V and W denote free D -modules of rank 4 and 5, respectively. Assume that
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

MA554_JAN03 - Math 554 (12) Qualifying Exam Heinzer January...

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