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Let $S=(s_1,s_2,...,s_m,...)$ be a linear recurring sequence with terms in$GF(q^n)$ and $T$ be a linear transformation of $GF(q^n)$ over $GF(q)$. Denote$T(S)=(T(s_1),T(s_2),...,T(s_m),...)$. In this paper, we first present counterexamples to show the main result in [A.M. Youssef and G. Gong, On linearcomplexity of sequences over $GF(2^n)$, Theoretical Computer Science,352(2006), 288-292] is not correct in general since Lemma 3 in that paper isincorrect. Then, we determine the minimal polynomial of $T(S)$ if the canonicalfactorization of the minimal polynomial of $S$ without multiple roots is knownand thus present the solution to the problem which was mainly considered in theabove paper but incorrectly solved. Additionally, as a special case, wedetermine the minimal polynomial of $T(S)$ if the minimal polynomial of $S$ isprimitive. Finally, we give an upper bound on the linear complexity of $T(S)$when $T$ exhausts all possible linear transformations of $GF(q^n)$ over$GF(q)$. This bound is tight in some cases.

The minimal polynomial of sequence obtained from componentwise linear transformation of linear recurring sequence