# 10-240-a6 - L-1 where L is from Exercise 1 iii Generalize...

This preview shows page 1. Sign up to view the full content.

MAT 240, University of Toronto Fall term, 2010 Exerzitien VI Due no later than 2:30 pm on Nov. 4, in your tutorial. 1 2 3 4 Exercise 1. Let L : R 5 -→ R 5 be the left-shift linear operator defined by L : ( x 1 , x 2 , x 3 , x 4 , x 5 ) 7→ ( x 2 , x 3 , x 4 , x 5 , 0) . i) What is the matrix of L using the standard basis for R 5 ? ii) Compute all L k for k = 1 , 2 , . . . . iii) Determine dim null( L k ) and dim range( L k ) for k = 1 , 2 , . . . . Exercise 2. Let S : U -→ V and T : V -→ U be linear maps such that ST and TS are isomorphisms from V -→ V and U -→ U , respectively. Prove that S and T are themselves isomorphisms. Exercise 3. Fix the matrix A = 1 1 0 1 . Define a map T : Mat(2 , 2 , Q ) -→ Mat(2 , 2 , Q ) via T ( X ) = AX - XA. i) Prove that T is a linear map. ii) Find a basis for null( T ) and range( T ). Exercise 4. i) Let S : W -→ W be a linear operator and suppose that S n = 0. Show that S - 1 is invertible, i.e. an isomorphism. Note: If c is a scalar, then S - c really means S - cI , where I is the identity operator. [Hint: since S n = 0, it is also true that 1 -
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: L-1, where L is from Exercise 1 . iii) Generalize the result in the following way: Show that if ( T-a ) n = 0 for some scalar a and operator T : W-→ W , then T-b is invertible (for a scalar b ) if and only if a 6 = b . [Hint: consider S = ( b-a )-1 ( T-a ).] 1 Reading suggestion: Read the rest of Axler Chapter 3 very carefully, and familiarize yourself with Chapter 4- it contains some basic results about polynomials in one variable. We won’t be going through it in great detail, but you should be comfortable working with polynomials. 2 I encourage you to work together on the ideas but the solutions must be individual. 3 Neatness counts, and so does being concise. 4 Please submit the questions in the correct order. 1...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern