6. Let V be the vector space above. Consider the maps T : V → V...

6. Let V be the vector space above. Consider the maps T : V → V And 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).

Expert Solution

Colby Messinger answered 2 years ago

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