problem2_sol - EE221A Problem Set 2 Solutions - Fall 2011...

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Unformatted text preview: EE221A Problem Set 2 Solutions - Fall 2011 Problem 1. Linearity. a) Linear: A ( u ( t ) + v ( t )) = u (- t ) + v (- t ) = A ( u ( t )) + A ( v ( t )) b) Linear: A ( u ( t ) + v ( t )) = t e- ( u ( t- ) + v ( t- )) d = t e- u ( t- ) d + t e- u ( t- ) d = A ( u ( t )) + A ( v ( t )) c) Linear: A ( a 1 s 2 + b 1 s + c 1 + a 2 s 2 + b 2 s + c 2 ) = s (( b 1 + b 2 ) t + ( a 1 + a 2 )) dt = = s ( b 1 t + a 1 ) dt + s ( b 2 t + a 2 ) dt = A ( a 1 s 2 + b 1 s + c 1 ) + A ( a 2 s 2 + b 2 s + c 2 ) Problem 2. Nullspace of linear maps. Assume that A : U V and that U is a vector space over the field F . N ( A ) := { x U : A ( x ) = v } . So by definition N ( A ) U . Let x,y N ( A ) and , F . Then A ( x + y ) = A ( x ) + A ( y ) = V + V = V . So N ( A ) is closed under linear combinations and is a subset of U , therefore it is a subspace of U ....
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problem2_sol - EE221A Problem Set 2 Solutions - Fall 2011...

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