sol2 - EE 221: Linear Systems HW #2 Solutions Sam Burden...

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EE 221 : Linear Systems HW #2 Solutions — Sam Burden — Sep 17, 2010 1 Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. (a) Linear: H ( a ( u + v )) = au ( - t ) + av ( - t ) = aH ( u ) + aH ( v ) (b) Linear: H ( a ( u + v )) = a R t 0 e - σ u ( t - σ ) + a R t 0 e - σ v ( t - σ ) = aH ( u ) + aH ( v ) Exercise 2. (a) Linear: H ( a ( X + Y )) = aAX + aXB + aAY + aY B = aH ( X ) + aH ( Y ) (b) Linear: H ( a ( X + Y )) = aAX + aBXC + aAY + aBY C = aH ( X ) + aH ( Y ) (c) Not: n = 1 = H ( x + y ) = Ax + Ay + B ( x + y ) 2 6 = Ax + Ay + Bx 2 + By 2 = H ( x )+ H ( y ) Exercise 3. Two cases: b / ∈ R ( A ) = S = ( 6 = { 0 } ), dim S undefined (0 or - 1 were also accepted) b ∈ R ( A ) then x : Ax = b and S = N := { x + ξ : ξ ∈ N ( A ) } , hence dim S = dim N ( A ): S N : ξ ∈ N ( A ) = A ( x + ξ ) = Ax = b S N : Az = b = A ( x - z ) = b - b = 0 = x - z = ξ
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