(Solution) - Show that linear independence is preserved by one to one transformations and -(2025 Original AI-Free Solution)

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Show that linear independence is preserved by one-to-one transformations and that spanning sets are preserved by onto transformations. More precisely, if T: V ? W is a linear transformation, show that:
(a) If T is one-to-one and (v1,..., vn] is independent in V, then (T(v1),..., T(vn)} is independent in W.
(b) If T is onto and V = span{v1 ..., vn], then W = span{T(v1),..., T(vn)}.