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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 0 0 y)|x, y ER}. B = {(x x + y x + y y)|x, y ER). C = {(x, y, z, w) | x, y, z, w ER and x + y + w = 3}. The set of functions D = {f: R rightarrow R | df/dx + 2f = o}. E = {f:R rightarrow RE | 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 0 0 y)|x, y ER}. B = {(x x + y x + y y)|x, y ER). C = {(x, y, z, w) | x, y, z, w ER and x + y + w = 3}. The set of functions D = {f: R rightarrow R | df/dx + 2f = o}. E = {f:R rightarrow RE | 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.