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Decide whether each set forms a vector space under the usual operations of addition and scalar multiplication. If the set does form a vector space, give a brief proof; if not, explain why not (Use the Subspace Lemma 2.9 if possible.) The diagonal matrices A ={( )|x,y R}. B ={( )|x,y R} C ={(x,y,z,w) | x,y,z,w R and x + y + w = 3}. The set of functions D ={f : R rightarrow R | df/dx + 2f = 0}. E ={f:R rightarrow R | f(7) = 0}. F ={f: R rightarrow R | degree(f) 2 or f(x) 0}. We write f (x) 0 for the zero function defined by f (x) = 0 for all x.
Show transcribed image text Decide whether each set forms a vector space under the usual operations of addition and scalar multiplication. If the set does form a vector space, give a brief proof; if not, explain why not (Use the Subspace Lemma 2.9 if possible.) The diagonal matrices A ={( )|x,y R}. B ={( )|x,y R} C ={(x,y,z,w) | x,y,z,w R and x + y + w = 3}. The set of functions D ={f : R rightarrow R | df/dx + 2f = 0}. E ={f:R rightarrow R | f(7) = 0}. F ={f: R rightarrow R | degree(f) 2 or f(x) 0}. We write f (x) 0 for the zero function defined by f (x) = 0 for all x.