Question: Let v1, . . . , vn be vectors in a vector space V and let T: V W be a linear transformation. (a)

Let v1, . . . , vn be vectors in a vector space V and let T: V → W be a linear transformation.
(a) If {T(v1) , . . . , T(vn)} is linearly independent in W, show that {v1, . . . , vn} is linearly independent in V.
(b) Show that the converse of part (a) is false. That is, it is not necessarily true that if {v1, . . . , vn} is linearly independent in V, then {T(v1), . . . , T(v)} is linearly independent in W. Illustrate this with an example T: R2 → R2.

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