(a) Let T : V W be a linear transformation, and suppose that dim[V] = n. If Ker(T ) =

Chapter 6, Problem 22

(choose chapter or problem)

(a) Let T : V W be a linear transformation, and suppose that dim[V] = n. If Ker(T ) = {0} and {v1, v2,..., vn} is any basis for V, prove that {T (v1), T (v2), . . . , T (vn)} is a basis for Rng(T ). (This fills in the missing details in the proof of Theorem 6.3.8.) (b) Show that the conclusion from part (a) fails to hold if Ker(T ) = {0}.

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