10-240-a6 - L-1, where L is from Exercise 1 . iii)...

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

View Full Document Right Arrow Icon
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 ² . 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 - S n = 1.] ii) Use the result to find the inverse of
Background image of page 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 wont 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

Ask a homework question - tutors are online