554-HW-5q

# 554-HW-5q - 5 Let T be the linear operator on R 3 defined...

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5 Let T be the linear operator on R 3 defined by T ( x,y,z ) = ( x,z,- 2 y- z ). Let f be the polynomial over R defined by f =- x 3 + 2. Find f ( T ). 5 Let A be an n × n diagonal matrix over the field F . Let f be the polynomial over F defined by f = ( x- A 11 ) ··· ( x- A nn ). What is the matrix f ( A )? 5 For a,b ∈ F a field and a 6 = 0 show that B = { 1 ,ax + b, ( ax + b ) 2 , ( ax + b ) 3 , ···} is a basis for F [ X ]. 5 If F is a field and h ∈ F [ X ] of degree ≥ 1 show that the mapping f 7→ f ( h ) is a one-to-one linear transformation of F [ X ] into F [ X ]. Show that this transformation is an isomorphism iff deg h = 1. 5 Use Lagrange Interpolation to find f such that deg f ≤ 3 satisfying f (- 1) =- 6; f (0) = 2; f (1) =- 2; f (2) = 6. 5 Let L be a linear functional on F [ X ] such that L ( fg ) = L ( f ) L ( g ) for all f,g ∈ F [ X ]. Show that either L = 0 or there is a t in F such that L ( f ) = f ( t ) for all f inF [ X ]....
View Full Document

## This document was uploaded on 01/25/2012.

Ask a homework question - tutors are online