Math121sol3 - Math 121 Linear Algebra and Applications Prof...

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Unformatted text preview: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 3 Posted: Tue. Oct. 30, 2007 Written by: Luca Candelori Exercise 1 (1.6/35) . (a) Let v + W be a typical element of V/W . Since v ∈ V , we can write it as v = a 1 u 1 + ... + a k u k + a k +1 u k +1 + ... + a n u n Since u 1 ,...u k are all in W , in V/W this becomes v + W = a k +1 u k +1 + ... + a n u n + W which shows that the set { u k +1 + W,...,u n + W } spans V/W . To prove linear idependence, suppose there are scalars a k +1 ,...,a n such that w = a k +1 u k +1 + ... + a n u n ∈ W then we can write w as a linear combination of the u 1 ,...,u k , since they form a basis for W . But this would violate linear independence of the u 1 ,...,u n , since they form a basis for V . (b) We know that dim W = k and dim V = n so that there are n- k elements in the basis for V/W provided. Therefore dim( V/W ) = dim V- dim W Exercise 2 (2.1/5) . We prove first T is linear. Suppose we have polynomials f,g ∈ P 2 ( R ). Then T ( f ( x ) + g ( x )) = x ( f ( x ) + g ( x )) + ( f ( x ) + g ( x )) = xf ( x ) + xg ( x ) + f ( x ) + g ( x ) = ( xf ( x ) + f ( x )) + ( xg ( x ) + g ( x )) = T ( f ( x )) + T ( g ( x )) Also, for any real number c , we have T ( cf ( x )) = xcf ( x ) + ( cf ( x )) = c ( xf ( x ) + f ( x ) ) = cT ( f ( x )) Write now f ( x ) = ax 2 + bx + c , with a,b,c ∈ R , a typical element of P 2 ( R ). For it to be in N ( T ) we need T ( f ( x )) = 0, or, in other words x (...
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This note was uploaded on 10/23/2010 for the course MATH Math 121 taught by Professor Bieri during the Fall '07 term at Harvard.

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Math121sol3 - Math 121 Linear Algebra and Applications Prof...

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