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hw3Jan2007

# hw3Jan2007 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 3...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 3 Posted Wednesday 24 January 2007. Due Wednesday 31 January 2007 in class. 1. [20 points] Consider the following sets. Demonstrate whether or not each is a vector space (with addition and scalar multiplication defined in the obvious way). (a) { x R 2 : x 2 2 = x 1 } (b) { x R 3 : 4 x 1 + x 2 + 2 x 3 = 0 } (c) { f C [0 , 1] : f (0) 0 } (d) { f C [0 , 1] : max x [0 , 1] f ( x ) 1 } (e) { f C 2 [0 , 1] : d 2 f/dx 2 = 0 for all x [0 , 1] } 2. [24 points] Recall that a function f : V W that maps a vector space V to a vector space W is a linear operator provided (1) f ( u + v ) = f ( u ) + f ( v ) for all u, v in V , and (2) f ( αv ) = αf ( v ) for all α R and v V . Demonstrate whether each of the following functions is a linear operator. (Show that both properties hold, or give an example showing that one of the properties must fail.) (a) f : R n R m , f ( u ) = Au for a fixed matrix A R m × n . (b) f : R n R m , f ( u ) = Au + b for a fixed matrix A R m × n and fixed vector b R m .

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