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 onetoone 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 ]....
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This document was uploaded on 01/25/2012.
 Spring '09

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