a2 - R 4 . (b) Show that the subspace in 3( a ) is equal to...

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McGill University Math 236: Algebra 2 Assignment 2: due Friday, February 3, 2006 1. Let V be a vector space over a field F and let v 1 ,v 2 ,v 3 ,v 4 ,w V . (a) Prove that if ( v 1 ,v 2 ,v 3 ,v 4 ) spans V then so does ( v 1 - v 2 ,v 2 - v 3 ,v 3 - v 4 ,v 4 ). (b) Prove that if ( v 1 ,v 2 ,v 3 ,v 4 ) is linearly independent then so is ( v 1 - v 2 ,v 2 - v 3 ,v 3 - v 4 ,v 4 ). (c) Show that ( v 1 - v 2 ,v 2 - v 3 ,v 3 - v 4 ,v 4 - v 1 ) is linearly dependent. (d) If ( v 1 ,v 2 ,v 3 ,v 4 ) is linearly independent and ( v 1 + w,v 2 + w,v 3 + w,v 4 + w ) is linearly dependent, show that w Span(v 1 , v 2 , v 3 , v 4 ) 2. Determine whether the given sequence of real vectors are linearly independent or not. In the case of linear dependence find a non-trivial dependence relation. (a) (1 , 1 , 1 , 1) , (1 , 1 , 2 , 2) , (2 , 2 , 1 , 1) , (1 , 1 , 3 , 3) , (4 , 4 , 3 , 3) R 4 . (b) f ( x ) = sin( x + 1) ,g ( x ) = sin( x + 2) ,h ( x ) = sin( x + 3) R R (c) f ( x ) = e x ,g ( x ) = xe 2 x ,h ( x ) = x 2 e 3 x R R (d) (1 , 2 , 3 ,...,n,. .. ) , (1 , 2 2 , 3 2 ,...,n 2 ,... ) , (1 , 2 3 , 3 3 ,...,n 3 ,... ) R 3. (a) Find a basis for the subspace of R 4 spanned by the vectors in 2( a ) and complete this basis to a basis of
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Unformatted text preview: R 4 . (b) Show that the subspace in 3( a ) is equal to Span((3 , 3 , 7 , 7) , (7 , 7 , 3 , 3)). 4. Let U = Span((2 , 1 , 1 , 3) , (1 , 2 , 2 , 1) , (3 , 2 , 1 , 5)), V = Span((1 , 4 , 3 , 5) , (1 , 1 , 2 , 2) , (4 , 3 , 1 , 7)). Find bases for U + V and U ∩ V . 5. Let U be the subspace of R ∞ defined by U = { x = ( x 1 ,x 2 ,...,x n ,... ) ∈ R ∞ | x n +4 = 5 x n +2-4 x n ( n ≥ 1) } . Prove that the infinite sequences x,y,z,w defined by x n = 1 , y n = (-1) n-1 , z n = 2 n-1 , w n = (-2) n-1 form a basis for U . You may assume as known that dim( U ) = 4. Use this to find a formula for the n-th term of the infinite sequence u ∈ U where u 1 = 1 ,u 2 = 2 ,u 3 = 3 ,u 4 = 4....
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This homework help was uploaded on 04/18/2008 for the course MATH 236 taught by Professor Toth during the Winter '06 term at McGill.

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