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can anyone tell me if I understand this right?

I have a t-invariant subspace with basis B, and I extend the basis B to be a basis B' for the entire vector space by adding L.I. vectors to it. Then I put B under a linear transformation, T:V --> V, and I will get a set of vectors in the range of T that generates W in that space, i.e. R(T) = span (T(B)). But since the subspace (let's call it v) of V is T-invariant, then the vectors I end up with in R(T) are also in the subspace (minus the ones we added to extend the basis). Is that correct?

Also, can something only be T-invariant if it's a mapping within the same vector space? Can subspace v of V be t-invariant if the transformation is T: V --> W? I'm not sure if that makes sense, because the generating set in the R(T) couldn't be generating the same subset as in V.