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**Unformatted text preview: **McGill University Math 270-2010 Fall: Applied Linear Algebra Written Assignment 3: due on Monday, Nov. 15, 2010, hand in in class. 1. Linear transformations: (a) Let P n be the space of polynomials of degree less than or equal to n in a variable x with real coefficients. Show that the following functions from P n into P n are linear transformations. • ( T ( f ))( x ) = f ( x + a ); • ( T ( f ))( x ) = a f + ( b + b 1 x ) f + ( c + c 1 x + c 2 x 2 ) f 00 , where a ,b ,b 1 ,c ,c 1 ,c 2 are real numbers. (b) Let P be space of all polynomials. Show that the following functions are not linear transformations from P to P . • T ( p ) = p + p + 1; • T ( p ) = p 2 . 2. Matrix representations (a) Find the matrix representations of the following linear operators on P 3 relative to the basis { 1 ,x,x 2 ,x 3 } . Let D : P 3 ⇒ P 2 be the differential operator, D ( f ) = f • T ( f ) = f + f ; • (T(f))(x)= f(x+1)....

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