6. Let V be the vector space above. Consider the maps T : V ? VAnd S : V ? V
defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...)= (0,a1,a2,...).
(a) [optional] Show that T and S are linear.(b) Show that T is surjective but not injective.(c) Show that S is injective but not surjective.
(d) Show that V = im(T) + ker(T) but im(T) ? ker(T) ?= {0}.
(e) Show that im(S) ? ker(S) = {0} but V ?= im(S) + ker(S).
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