problemset9 - MAS 213: Linear Algebra II. Problem list for...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
MAS 213: Linear Algebra II. Problem list for Week #9. Tutorial on 9th November. This week’s topics: The eigenvalues and eigenvectors of an abstract linear operator. The multiplicity of an eigenvalue, and how it compares to the dimen- sion of the corresponding eigenspace. Diagonalizability for matrices (continued). Tutorial problems: Problem 1: (Problem 5.1.4, various parts, from [FIS].) In the parts below you are given a vector space V and a linear operator T : V V . Determine the eigenvalues of T , and find an ordered basis of V with respect to which the operator has a diagonal matrix representation. (i) V = P 1 ( R ) and T ( ax + b ) = ( - 6 a + 2 b ) x + ( - 6 a + b ). (ii) V = M 2 × 2 ( R ) and T ( A ) = A t + 2 · tr( A ) · I 2 . Problem 2: (Problem 5.2.7 from [FIS].) Let A = ( 1 4 2 3 ) M 2 × 2 ( R ) . Derive a formula for A n , where n N . Problem 3: Let A be a diagonalizable matrix. Show that every eigenvalue is either 1 or 0 if and only if A 2 = A . Hint: In this problem and the next one, you should use the theorem that
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/08/2010 for the course SPMS MAS213 taught by Professor Andrewkricker during the Fall '10 term at Nanyang Technological University.

Page1 / 3

problemset9 - MAS 213: Linear Algebra II. Problem list for...

This preview shows document pages 1 - 2. Sign up to view the full document.

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